So who's going to fund this adjudicatory system? Because lots of little $10 deposits, that are fully refundable, won't do it.

You should ask Michael Saylor to be sure how he proposes this will all work, but he did mention his company (MicroStrategy) spends $1M/year on DDOS mitigation. I expect if Lightning ends up as cheap and fast as he and others are saying it will be, it could be funded by companies as a cost of doing business in the same way they fund the electricity and hardware for serving their websites.

It's actually the sort of thing I could see CloudFlare etc. providing, so each website doesn't need to implement it themselves.

Who decides whether your request is abusive?

The site. They stake their reputation on being fair. Unfair sites will lose users. This is basically what happens already with moderation, it's just adding a small monetary penalty to bad actors to discourage them.