If it's cPanel-to-cPanel, and your host is willing to assist in the transfer, and you're willing to give them the cPanel password to your old site, and the data capacity to be transferred is within a reasonable size, they may be able to do a direct transfer of everything from server-to-server.
You need to talk to them about this. It's not something you can do yourself.
I don't understand your template question ... a template is a template. You can install a template on your new host the same as you would on the old one.
As to databases, if you keep the same password, that's one less thing to make a mistake on. But if you use a different one, you have better security since you can start fresh. And, if your host is transferring cPanel-to-cPanel for you, they'll take care of the database automatically, because it'll recreate the old one on the new server for you exactly as it was before.
Again, talk to your hosting company.