If we are talking about the databases, then I have the most experience with Neo4j. I tried some of the other DBs (orientDB, Janus, Neptune), but only for pet projects. So that is not so relevant.
I think for hosting options you should consider your requirement/architecture/deployment/budget setup and choose the best fit for that. This should not be a graph database specific question, this is more general. We usually use AWS EC2 if it is a cloud setup, but GCP is popular too.
Some hints can be found here: