A fully managed service should help you with pretty much anything except for 3rd party related issues (since no one can guarantee help on any 3rd party issue, on things out of the host's control, such as a custom coded script or plugin), but its always good to go with a managed host that at least tries to resolve the 3rd party issue.

A fully managed solution will indeed cover everything apart from applications you install/configure on the server

It's important that the company understands your needs. Everyone is different...

I find smaller companies are a lot more personable because they value every client, as opposed some of the larger giants like Rackspace. This isn't to say they don't value every client but the impact is much less direct if a single client leaves.