If I buy an app from the app store, do I have to stay logged into the app store?
Yes, although there are many questions that cover the same ground already on SE.

You can DL app store apps (including Lion) on your girlfriend's computer. Then she can log into her own apple account and buy other apps. The only thing you need to be aware of is that for updates, you can only update the apps you bought with that particular account. So if you want to update Lion (not likely as it will come through software update) or other apps, you will have to re-log into your account.

I believe most apps have a 5 computer limit.