This might be a bit late but I got an S3 in late june. Firstly you are going to want to install some custom firmware, i recomend cyanogen mod if you've never done this before. After you've done that or if you decide not to then you need to get some nice third party apps to replace some of the nasty samsung/google ones. PowerAMP is probably the best music player out there if you're willing to pay a small price and MX player for video. Next you're going to want a 3rd party keyboard Swiftkey 3 is a godsend and i highly recomend it to anyone using an android device that supports it. Quickpic is a really nice free gallery replacement that adds lots of nice features. You need a file manager like Astro which is also free. In terms of games Fruit ninja, world of goo, angry birds, racing game of choice, tower defence game of choice and physics games. Although allot of these apps are premium a.k.a. have a price tag, they are also available from certain places free if that's what you're into. LAST BUT NOT LEAST GET A DAMN CASE AND NEVER TASKE IT OFF! I DROPPED MY S3 2 WEEKS AGO WITHOUT THE CASE ON AND CRACKED THE SCREEN!!!! D:
Hope this helps.
You won't need to root it for the official update, it may just not be available from your carrier yet. If you wanted to install a custom ROM on the other hand you would need to root the phone although rooting and installing custom ROMs voids the manufacturers warrenty it does improve performance of the phone due to the lack of bloatware etc.
Best thing to do is just wait it out, im sure the update will be out soon.
Not really, but a smaller case that's just as protective would be nice(if it even exists). It's response time has gotten slower. When I try to reject/answer calls with a swipe or even unlock the phone it can take multiple swipes to do it whereas it use to take only one.
That is probably your issue, I'd say maybe get the otter box commuter and a pack of screen protectors from amazon. I got 10 screen protectors for like £2 and they do the job nicely. Try your phone without the case first to see if the input lag is caused by your case then go from there, if it isn't the case then you are gonna want to ring your carrier.