Works like a charm, as if I had a new phone! There are no overheating issues anymore and the battery can last up to 2 days. I was thinking of installing Cyanogenmod, but their current stable ROM is based on Android 4.0.x, so I opted for the original and new.
I highly doubt it. It’s already official that Galaxy Note 1 (from the same era as S3) won’t receive 4.2.2, let alone 4.4. I would try CyanogenMod on your place.