I've never had a prime shipment miss the date given at checkout. But I have had the guaranteed dates go out way more (like 4-5) than 2 days. The last time I started a chat window to ask why stuff I was ordering on Thursday was going to take until the next Wednesday, and they upgraded it overnight shipping for free.
It seems to happen more lately, I also wish I understood what stuff will deliver on the weekends and what won't.
As far as sellers go. The only third-party sellers that ship with prime are the "Fulfilled by Amazon" sellers. You can read all about it on amazon's page. But those sellers pay Amazon a fee to store the item, handle the item and ship the item, as well as the typical commission. I am sure that is a pretty standard arrangement with any fulfillment center. Sellers that are not fulfilled by Amazon can charge whatever they want for shipping and that goes directly to them, but it isn't shipped via prime.
I will say I've never had a problem with Amazon customer service. I think they typically go way beyond what is expected, which is why I pretty much never give my money to Newegg anymore.
When the UPS -> USPS handoff first started, it was horrible, took days. I haven't had a problem with in since then, and I get a lot of prime orders through USPS.