For the shield delivery invoices and other gear you don't want, the best thing to do is go to the Crafter. There is one in each hub, in Avengers Tower it is Hank Pym, go to the right of the portal / central area, through the glass doors, then down to the right. Instead of selling the items, alt + right click to "donate" them instead. This levels up your reputation with the crafter and unlocks new crafting recipes.
Do save the 5 crafting mats and the Matrix of Unbinding. They all stack and are needed for endgame recipes if you like the game enough to keep playing.
Yes, save the cupcakes for later. They are free boosts that will still be usable after the anniversary ends at midnight on July 4.
You can donates all the rings, medallions, insignias that you aren't using. They are easily replaced.
The giant chest of drawers to the right of the portal is your bank space (S.T.A.S.H.) -- I'm not sure how much space new players get now since I first played back in 2014.
Things to consider saving besides the crafting mats are "artifacts", "uniques", "legendaries", costumes. For artifacts you can compare two of them to see which one has the higher numbers then donate the other.
If you want more STASH space, through Monday you can get extra tabs of them for $3.50. You can also save up 365 cake slices to get a stash tab, though if you only have enough for one thing the hero token is worth more.
Also consider the $2.49 Civil War DLC on Steam to unlock Captain America and Iron Man, plus get extra costumes for them and hero-specific-gear STASH tabs for them.
For gear, besides donating you can also feed them to Lucky to level up the pup. Once he's eaten enough of each rarity of gear he gives you a bonus like 8% more health or 2% less damage from bosses. You can feed Lucky by hand or using the J vacuum command or by using ESCape > Options > Gameplay and using the sliders for gear slots 1-5 to turn on "vaporizing" of that rarity. You should be able to set it to vaporize at least Rare (blue) gear so you only ever see purple (epic), yellow (cosmic), gold-brown (unique) gear. Lucky chows down on the vaporized gear.