I wouldn't necessarily install Linux on the portable hard drive and boot it from there. If you want to be able to boot Linux over USB you could just get a 4 or 8 gb thumb drive and make it bootable and install Linux. That way you won't have to deal with your computer trying to load Linux every time your external drive is connected even if you don't need it to.
But more to your questions: You should have no problem partitioning the hard drive and almost assuredly you could run Linux off of it. When you do your partitioning you'll need to flag the partition on which you're installing Linux as bootable and for your data partition you will probably want to make sure you format it as either FAT32 or NTFS so that it can be mounted in both Windows and Linux. FAT32 would probably be better if you are using it just for backups because it is interoperable with nearly every operating system.
I don't have a specific recommendation for drives, hopefully someone else can help there. Before buying one from a store like Staples at least double check online at a site like
Newegg.com - Computer Parts, Laptops, Electronics, and More! to get reviews.