Can Bitcoin be "shutdown" if energy consumption for mining is heavily regulated? Can Bitcoin be "shutdown" or heavily hampered if energy consumption for mining is heavily regulated? Can transactions still be processed if mining for new coins is hampered? I am thinking it would be easy to detect illegal mining just like they detect illegal drug-plant farms from the massive use of energy used to power the solar lights. Would BTC survive with only minor mining going on?

It is quite obvious that if BTC will start to consume more energy than USA than people might be, if nothing else, a bit concerned about the environmental effects.

If BTC won't survive, has Ethereum with the future proof-of-stake concept a better chance of surviving government clamp-down?
Those questions have been answered multiple times.

The difficulty depends on the mining power. If there is a regulation and 60% of the miners are away, the difficulty would decrease drasticly within 2 weeks and the other 40% would need less power to mines the blocks. So the mining would not be hampered for a long time.


  Would BTC survive with only minor mining going on?


yes. Only a few people could still mine. The bitcoin network does not really need big mining-pool companies for that.

And you third question is a primarily opinion-based.