You seem to have misunderstood "mere accusations."
Takedowns due to "mere accusations" to me means that I can fire off an e-mail if I see a kickstarter project that looks sketchy and have it taken down post-haste.
Well if it's based on this: http://news.ycombinator.com/item?id=4549792 then Kickstarter should take it down. It's a civil matter, so there's no "beyond reasonable doubt" required, and if Kickstarter are helping fund infringement, then they're on the hook too.
Is there actually a legal basis for this? As in, if Stripe was, in good faith, processing payments for a company which is selling (allegedly) infringing products, they're obligated to stop processing payments immediately? It seems like there'd be a clause in the TOS to indemnify them.
They didn't just take it down - they left no way for anyone to find out what had happened to the project by looking at the Kickstarter site. There's just a page saying the project doesn't exist anymore for some unspecified reason.
Yes, it should - if it's backed by a legal letter, Kickstarter might be liable too, if the suit is sucessful.