Ummm, I'll give it a shot. I've got the Verizon Tempered Screen Protector, which is ok. Some times I have to press once, twice, or even three times to get it to work, but it hasn't been annoying enough to make me take it off yet.
As for a case, I ordered the Tech21. Highly recommend it. It does not add that much extra bulk. Yes, I'm happy as I can be with my LG G3. Could it be better? Well, sure, but is it a great device, I think so.
Pictures of my phone with the Tech21 case and you can see the screen protector as well. However, you really cannot tell the screen protector is on. It's not as noticeable and the pictures would indicate.