Yes, when you sign with one of these servers they give you full access to the node. From there on you just install stuff as it was your computer.

Now, abou me setting one up... Maaaan, I'm thinking about setting up a BTC node for a couple of years already and maybe now is the time to do it.

And, answering your question, yes... If you set your node on one of the hosting providers that they describe on the documentation, you then just need to point a DNS record there to have it under your own domain.