Home button doesn't click properly

The missing home button gasket is most likely your issue. Definitely replace it, but while we’re in there, check the bracket itself and make sure that the little raised piece on the bracket is still in tact. This is what the actual click mechanism on the home button pushes into to make the click. I’ve attached a picture and circled the area I am referring to. I’ve seen a couple of times where this piece broke/fell off so there was nothing to press into the click mechanism on the home button itself. Doesn’t hurt to take a look while you’ve got it off.

As you’ve already guessed, the missing gasket is most likely the problem. When adhered to the screen properly, it allows the button to “spring back” towards the screen after you’ve pressed it. Without it, with every button press, you’re sort of brute force pushing the button into the bracket, which could account for the screws working themselves loose. You want to get the screws “finger tight”. Don’t torque them down with any sort of force, but just enough you can do with your fingers alone.

Let us know how it went after you’ve replaced the gasket!