I just went through something similar in a test environment so here's my comments.
When doing the GP install, you can have it create the ODBC for you. I don't believe there is anything you need to download.
Yes, you can backup on old and then restore on the new server. You should be able to do everything without installing GP 2010 on the new server (although I did because we have several 3rd parties that i wanted to make sure were all installed and set up properly before beginning the upgrade).
I'm not sure what build of GP 2015 you were considering going to but I would go for one of the most recent hotfixes to 2015 R2. I started at just 2015 R2 and ran into an error during the upgrade that was fixed on a later hotfix. See below for some blogs regarding known issues:
community.dynamics.com/.../microsoft-dynamics-gp-2015-r2-known-upgrade-issues
I'd also suggest reading through the upgrade.pdf file that you can grab on customersource from the same area where you can download the software.
