Barb - You can solo the game with decent items, but you will get trashed in PvP unless your the new BvC (Barb vs Caster) variant which has 6k+ life and can take one hell of a beating. Even then your probably gonna have an easier time with a pally or druid.
Player Vs. Player in D2, in my humble opinion, is a joke. I will agree with you however that the Barbarian is generally (not always, depending on the gear, of course, but also depending on which skills were maxed out) weak in PvP, but then again, I myself only played PvP in a very non-serious level back in the 1.08 to 1.09 days. On the other hand, the Barbarian in Player Vs. Monster is in my opinion the best there is to solo the game, but yes that's just me, I'm a Barbarian fan, I have one right now at level 93 in Ladder and 94 in Non-Ladder.
My N-L and Ladder Barbarians, a Mace Barb and a Sword Barb respectively have up to 95+ in Resistances in Hell, deal elemental, physical and magic damage with up to 1.8K to 2.8K damage completely excluding bonuses from Crushing Blow, Deadly Strike and Critical Hit. They can eat Stygian Dolls and Souls and even Oblivion Knights casting Iron Maiden on them for breakfast, and in grave danger they can Howl their way out of the crowds when Whirlwind isn't an option due to Iron Maiden or other possible reasons. The Physical Immune monsters means no threat at all thanks to Berserk, and if dexterity is lacking and chances to hit certain monsters in Hell is low they can cast Battle Cry to not only reduce their target's defenses by up to 90% but reduce their physical damage by up to 50%.
All that and more including 250%+ Magic Find excluding Find Item skill which adds around another extra 50%MF with the + to Skills items I have. I don't even need any Merc to comfortably go solo in Hell. And so on and on, without even mentioning the Merc I have gives me a Might Aura, and I too have one as well along with Sanctuary Aura which knocks back the Undead, making a Chaos Sanctuary run a pleasure to do. Yes, to me the Barbarian is the best class in Diablo II, but I've been making a couple of dozens of builds with them since 1.08, I'm very used to them, my next most played build being the Amazon and the Necromancer.
Necro - Probably cant solo the game, but with enigma (the teleport armor) your good in PvP, they are no longer the joke they were in 1.09.
The Necromancer in 1.10/1.11/1.12 can solo Hell mostly if he's a Summoner. The Summons and also the Revives can be tremendously powerful, along with the Iron Golem or even Blood Golem, and let's not forget the Merc's help too which in the end with good gear can always provide an Aura to the party passively. Also the Necromancer's Amplify Damage curse can remove certain immunities such as PI's (Physical Immunes) and others, not everywhere, not against all monsters, but a great number and varieties of them are affected by that curse's advantage.
The last Necromancer I made was in Ladder Season 3, he was a Summoner and I barely ever joined any public or private party games to gain the advantage of a rush through Hell or any help for Quests, I did everything alone including my Magic Find runs and my Experience runs (Baal runs and Chaos Sanctuary runs). The initial difficulty for a Summoner/Reviver resides only into actually finding the proper monsters to Revive for the best effectiveness and the best Skills attribution and items used for the best results with his Summons. When that is done and when he knows where to go (mostly, but not always since it's often random) for his "good" Revives then such a Necro build can easily, yes easily do Hell alone.
On the other hand, a Poison and Bone Necromancer will have a hard time without very good gear, and even then... I'm not sure.
Pally - The new best class imo, can solo the game, can magic find, and can trash most other chars in PvP, its what the sorc used to be.
Without Enigma the Paladin suddenly becomes quite a generic class, sometimes comparable to the Barbarian, but not as effective, not at all. As I said, remove just one Skill from him and one, ONE item in the game, and oh... suddenly he's not as powerful. The Skill I'm referring to is Blessed Hammer (Hammerdin builds), and the item, already mentioned, the Enigma RuneWord. Remove both and you suddenly can't fight when Iron Maiden is cast upon you, and you suddenly cannot even Smite your way alone in Hell (in other words even a good Smiter build cannot entirely solo up to Hell without encountering great danger on his way, I'd love to see a Smiter alone in Chaos Sanctuary without Enigma and Blessed Hammer, good luck, the easy part will be Diablo, but only him and nothing else).
Yes, the Hammerdin is extremely powerful, too powerful in fact. Just one Skill, its Synergies, and one single item in the game makes him so efficient (simply because Hammerdin deals Magical Damage like nothing else in the game, with the only competition in terms of efficiency being the Barbarian's Berserk). Yes of course a good Hammerdin also needs other RuneWords to work at its full potential, but you get the point. Give one a good Exile or a good Herald of Zakarum shield, a good Call to Arms Flail and you're suddenly much better than what a Sorceress should be doing instead and you suddenly are the most beloved class build in the entire D2 community since 1.10 was released.
To me, the actual item/RuneWord Enigma and the introduction of Synergies giving that much advantages to the Blessed Hammer is the imbalance of Diablo II at the moment, much to the pleasure of the ones owning that build, of course.
Sorceress - You cannot solo the game, no way, in hell there are immunities to fire/lightning/cold everywhere and in PvP people can simply pull out absorb items, so you may get a kill but when they go back to their stash and come out with the absorb fire ammy/absorb cold shield your stuffed.
True, but the Sorceress has never truly been used as a solo "assaulter" class. In its nature Diablo II is a team game, yes. And the Sorceress just like the Paladin (prior to the discovery of and creation of the Hammerdin build) were meant as Party-Support classes, especially the Paladin with his Auras, but yes the Sorceress too. The main use for the Sorceress now and back then when I started during the 1.08 days was to do very predictable/set Magic Find runs to the places you know don't usually (or always don't) spawn monsters that are all Immune to your chosen Skill tree (Lightning, Cold or Fire). And if Sorceresses go with Hybrid builds (all three Skills trees, or even two of them) then their efficiency diminishes due to the lack of properly chosen Synergy-giving Skills.
Even the "best" Sorceresses I've seen this Season so far still are only Magic Finders, or characters Rushing others through Hell (but then again that might be dangerous depending on which monsters spawn and which Immunities they have, sometimes requiring the Rushing Sorc to ask the player who's being rushed to create a new game so to possibly give a better chance to the Sorc with perhaps weaker monsters spawns). But generally speaking, that's fine. The Sorc was never meant to be a solo'er.
Amazon - You dont see many zons around anymore, might be able to solo the game, not too good in PvP these days without the bug they used to have in 1.09. Still a good zon should still be a threat, unless your a teleporting paladin/druid.
The best Amazon builds, in my opinion (and from what I've tried myself) are Javazons, but Bowazons can be quite effective, mostly with the Faith RuneWord (there again, the example of what just a single item can change so many things in terms of what a class can now do which was barely possible before without it).
I do know right now that the #2 or #3 Amazon in this current Ladder Season simply rapped Hell Baal in an 8-Players game before most players following after that Baal Run actually reached Baal to fight him with that Amazon. It took her between 6 to 7 seconds to bring him down, which is faster than many good Hammerdin builds I know of and regularly do Baal Runs with. She's a Javazon build and I do not know her exact gear, but she probably had an Enigma. I was one of the players who didn't have time to get to Baal to see her fight him before he died during that Run. The others commented on it and were all quite surprised.
That build might be an exception, but generally speaking the Amazon is capable of doing solo in Hell, but she probably needs the help of an Act 2 Merc (giving perhaps Holy Freeze Aura or maybe Blessed Aim for added Attack Rating).
Druid - Nearly as good as the pally from what i can tell, it can magic find, it can also trash other chars in PvP, druid vs pally i dont know who would win, they both use overpowered area of effect skills in PvE and PvP so the first to attack would probably lose.
That's the only class on which I cannot comment much, I never touched that class, but from what I'm seeing right now in this Season, and from what I remember of past Seasons, the Elemental Druid seems to be the preferred choice between that build and the Shape Shifting build. I can also safely say that I haven't seen a single Druid going about without Hurricane, so that Skill must be powerful or at least one of the most efficient ones. They seem to be good Party-Support characters too, with their Oak Sage and their Wolves and his Bear coming to his aid. I may try one this Season.
Assassin - Not a clue, the only class i dont know about in 1.12. They were ok back in 1.09 but a lots changed since then so no idea, a teleporting trapsin may prove quite dangerous but thats just speculating.
I can comment on that one easily.
Right now in this Season I had the chance (and I should say honor as well) of knowing the #2 Assassin, named Cherylin (not the account name, but the character's name). She's a Trapasin, level 96 I believe, or if he reached level 97 recently it wouldn't surprise be, the guy barely ever sleeps to achieve it, but he worked his arse off to get where he is.
With that said, his build doesn't even have the required good gear to make his build "godly", and already he's quite devastating with her. And he's not only using Traps, but also various Shadow Disciplines skills such as Mind Blast which is the very best crowd control Skill I've ever seen in action since the Barbarian's Whirlwind. He got the Shadow Warrior around as well, along with an Act II Merc, but the last time I've seen the gear his using it wasn't all that great, so I wouldn't even imagine how great it could be with good gear.
He told me however that just like the Sorceress he often has to check out for Immunities (to Lightning especially, if not exclusively). But generally speaking to a lesser extent than the Sorceress since the Assassin, even a Trapasin build, can deal significant physical damage. He says his current build can definitely solo Hell, but he's got to be careful at times, he also said that he can rush people in Hell thanks to Mind Blast mostly, but the main Trapasin's force is to work in team, to put it in his words. There's very few truly Party-Independent classes (or I should rather say Builds) in Diablo II, other than the Barbarian and the Necromancer.