So I downloaded a few apps on my phone earlier but when I just synced up my iPhone with my computer the apps did not show up in my app library. What is the issue?
The easiest thing to do is to turn on automatic downloading in Preferences -> Store. Then whenever you purchase an app on the phone it will be automatically added to your iTunes library.
Nope. You don't have to sync anything. As long as you are signed into iTunes with the same ID that you purchased the app on your phone, then it should work fine. Even if you aren't signed in to iTunes, the next time you do, it should download the apps. It is not in any way tied to your device, just your Apple ID.