I have several iPods ranging from 1st and 2nd gen Mini to 5th and 6th gen Classics. I always use "Manually manage music" and have no trouble connecting my iPods to any of my 3 computers. It lets me connect to iTunes and add music from any of the 3 computers to the iPods.
As I understand it, the iPod is supposed to be a 'one-way' street for music. You can add music, but not get it off. You're not really supposed to be using your iPod to move music from one computer to another.
If your iPod is set to 'Enable disk use,' you can use it as a removable disk drive. It should show in My Computer and you can copy files to it like it was a USB thumb drive, then connect to the other computer an copy them off. When they're on the other computer, you can just add them to iTunes.
When it comes to which iPod to get and also just wanting to play music and have a good amount of space, I understand what you mean. I think an iPod Mini that's had its 4gb microdrive replaced with a Compact Flash card is a GREAT option.
They can use most of the common sizes of CF cards and they work GREAT. I've done several of them and feel the best value is the 32gb size, but 16gb, 8gb and 4gb would work. 32gb still a ton of space in a small package and will have plenty of room for four or five thousand of your favorite songs. (and I'm only adding this as a direct response to what iPod to get, but it just happens that I have a 32gb Mini listed on ebay right now.)
For me now, a 32gb Mini is my every day carry iPods, then my iPhone 5 gets to do everything else.
BW