In the first two cases, you are talking about your own property. Unless you have a contract with the colocation facility that says otherwise, the presence of your machine at such a facility shouldn't somehow make it searchable, just like if I put a closed briefcase full of papers down in a public library and walk away from it for 5 minutes, the police cannot suddenly search that.
As I understand it, things are shakier in case (3) and (4), because your provider absent a contract with you that says otherwise could choose to comply with subpoenas (providers commonly claim that they'll respond to LEOs only if they have a "subpoena or a warrant", as if that was a feature; you should assume that subpoenas are mere formalities and that any LEO can obtain one practically on demand). It's possible that your remedy in such a case would be civil, which would be cold comfort if your goal was to have evidence excluded from a trial.
There are a bunch of lawyers on HN now, and I'm sure one of them will jump in here to correct me.