Topic: Two Issues With PS3

I've recently got my hands on a PS3 and there are a couple of issues I'm hoping someone can help with.

First, the console had no controllers, so I bought a genuine Sony Dual Shock one from a well known buy/sell/exchange chain, but it seems overly sensitive and buttons seem to 'stick' without actually doing so. Is this definitely a controller issue, or is there any chance this could be a console issue?

Secondly, I'm having issues with bluetooth headphones. I've tried two pairs or relatively cheap ear buds bought on Amazon. While both pairs pair with the console, one pair remains silent, while the other pair picks up what I can only describe as 'ZX Spectrum loading a game' noise.

Can anyone offer any advice for either (or hopefully both) of my problems?

Thanks for reading!