If I acquire, for example, an Amulet of Life protection when I'm level 2 when its modifier is +1, does its modifier increase to +2 automatically when I reach level 7, or is that counted as a completely different item which I then have to "refind"? Is the answer the same for all other items and weapons in the game?
No. If you have a Amulet of Protection +1, it stays an Amulet of Protection +1. You will have to find, buy, or make an Amulet of Protection +2 if you want that instead.
All items work this way. Although there are artifacts (such as the Staff of Fraz-Urb'luu) that can increase their enhancement based on other parameters, they are the exception rather than the norm.