Depends what you mean by own the hardware. In certain situations we do buy the hardware and then co-locate. But that really isn't a big difference. Hosters don't even have our root password to our servers so they need to ask us when they go in.
It'll be pretty extreme if they load into recovery mode to reset the root password and in those cases
1) It MAY be a breach of laws to even do that.
2) The server will have to go offline for them to reset the root password and we have systems in place to alert us when a server goes down.
3) Once server goes down and we can't get back in cause of the root password reset then kinda obvious that something happened.
If they intercept incoming data, it's encrypted. The rest of this I will pm you.
I somehow think it's pretty unlikely that will happen since it's so extreme and would be a real nightmare for them. If they get a court order, the logical response would be to just comply and let them seize the server rather than try to use it to gather evidence and trap people.