I am curious how they’re funded. How they are able to stay online. Surely there must be people, governments etc with deep pockets that would want to take them down?

You can donate to get access to faster download mirrors. I'd guess this is the main source of their revenue.

https://annas-archive.org/donate

Can you donate to them without someone claiming you're donating money to a criminal enterprise and getting you in trouble? I mean, without using bitcoins

You can buy Amazon gift cards using bitcoin lighting to add another layer (actually 2) of paranoia :)

I suppose it could also be their enterprise users, though there’s not a lot of info on this aspect of their activity.

Allegedly, some companies with deep pockets have paid them for access to their collection. The collection turns out to be useful for training LLMs.

Can confirm this is happening. But the money paid is tiny. Think thousands of dollars, not millions. Not enough to keep the lights on. I would assume they do pretty well from donations.

Never assume anyone does well from donations. That's rarely the case.

Donors get benefits.

Source on this claim? All their torrents are released publicly. Why would "companies with deep pockets" need to pay them?

Because they have an interest in the ongoing work of archiving of new things I guess

For fast access

Paying someone, even a pittance, gives you deniability and a chain of ownership.

Using a torrent of the exact same thing does not.