Bitcoin mining rigs can only perform SHA256 hashes. So, alternative use cases are limited. They do give off lots of heat, so you can use them as noisy heaters

Yes, Bitcoin is much more than payment processing. It is also a very important store of value, a monetary unit not controlled by any government. The Bitcoin network is also the first working proof of decentralized cooperation and consensus among adversarial parties, aka Nakamoto Consensus. This could have many implications over time.

You can include small amounts of arbitrary data in every transaction and block. For example, miners can "sign their name" on a block they create, and/or include a brief political statement such as "NO2X". Something else you can do is include a hash of something, for example the hash of a statement or contact. This hash can then be referenced later as proof that the given statement/contract existed at the time it was included in a block.