Mine does this too. I have to switch both the console and controller off and then it works again for some reason...

My major problem is that the right trigger on my controller literally jammed for around a week and then became relatively normal again. However now when it's fully depressed I get a hard 'clunk' instead of the soft rubber feeling that the left one has. I think something broke inside.

This happens with every controller type, sadly. Happened for me with a DS1, DS2, and X360 controller as well, and a couple of Logitech ones to. Not out of the box, but in time. Sadly, the only workaround I got, was to simply change them...

The grove the circuit board fits in, might of snapped/popped out. Or the ribbon cable dislodged. When you push down you'd hear a plastic click noise. For both issues. Opening it up. And putting them back on, would fix it.

This used to happen to me with the c stick (yellow stick) on the GC wired controllers. The B button kept getting stuck also on multiple controllers. I kept sending them back to Nintendo and eventually I gave up. I bought a Wavebird controller and never had problems with buttons sticking or c stick not being centered (would be obvious in Smash because screen would tilt). So I bought 4 Wavebirds and never looked back.

Perhaps Microsoft wants you to upgrade to the Elite.

I'm just worried I won't be able to get it back together. I've never opened a modern controller. Is there any chance of parts popping out that might not go back in so easy?

No. The triggers are a pain in the ass to put back. But it's not hard. Best way to avoid goof ups. Take photos after each time you unscrew something. Then look back, if you forget.

I think if you slightly move the d pad or directional stick on turning on the console / controller this happens,

I had 3 XB 360 controllers with drift problems on the right stick. I wouldn't effect all games, but games with sensitive cameras could be problematic. I could not figure out how 3 controllers could have the same problem. It still makes no sense, and I never did find out what the problem was.

Never had issues with an Xbox controller but yes, that can happen. Had probably two or three 'defective' Dual Shock 2 back then.

Edit: Shouldn't be an issue with a small deadzone in the game.