Hosting Migration Question

Right now I have wordpress installed on a friend’s server, who also provides my hosting. However, that install has some issues, such as that the flash uploader doesn’t work, the plugin/theme installer doesn’t work, etc. I am moving to a new host this weekend (bluehost) and I was wondering if it is possible to use a new install (the auto-install) while still keeping my existing database? And if that is possible, would my old problems end up still occurring on the new host as well?

I have looked at the Moving WordPress FAQ, but I can’t find anything about this sort of thing on there.