"Can the proof-of-work from the mining be transmitted entirely over Tor?"

Yes. I'm pretty sure it's possible just like with bitcoin.

"what's stopping spammers from doing this?"

I think the main reason is because it costs too much in resources to be effective

"How large of a server farm is needed to mine $25"

I recently read that people were getting 1 or 2 SP per day by mining so thats around 10 - 20 cents per day mining with an average desktop cpu. I might be wrong about all of this though.