Both VPS and Dedicated servers are segmented into managed or unmanaged categories. Managed is where their tech team provides tech support. I'm not tech savvy enough to comment as to if https would help you any (wouldn't think so) if you just use your server for trading. My experience is limited to using servers to power websites. It's unclear to me exactly what you are exposing on the server. If it has your brokerage username / password, then just assume the worst and do your due diligence in seeking a provider others trust for like services. Not a comfort, Tech support for servers can be from any corner of the planet.
If you buy unmanaged, it's doubtful the seller will have a desire to poke into what you are doing unless you request help. Of course, that's probably true for managed as well. Reviews are all over the place, even with the best providers, you are bound to find a few unhappy campers. Even so reviews are a good indicator of overall service.