Congrats on getting the a100! It really is a pretty good tablet with only a couple of drawbacks. The screen viewing angles are good from 3 sides but terrible from the 4th side, and if you got the 8GB version, it doesn't leave you with a ton of storage if you like to play games that require large downloads. Still, both issues are manageable, so I think you made a good purchase.
Apps:
Circle Launcher - I like to keep my main home screen clean so I use this app to keep all of my most used apps out of sight. You can shrink the widget down to nothing so that it looks like there's nothing on the screen, but when you click on it, it pulls up your apps. I use it on my phone and tab.
MyAppSharer - Great way to send apps from my phone to my tab and vice-versa.
Rainy Days - Gives you a quick, detailed look at Doppler Radar.
SwiftKey Tablet X - We all have our own favorite keyboards and SwiftKey is mine. The prediction engine is uncanny. According to the stats, it has saved me 7,019 keystrokes since I installed it Jan. 4.
Astro File Manager - Allows you to delete stored apk's and anything else that using up that valuable storage space.
And of course Netflix, Nook or Google Books, Google Music, and Pulse.
Games:
It depends on what kind of games you like, but here are a few of the ones I like:
Temple Run - At first, I thought it was kind of lame and wondered what all the fuss was about, but after a few games it was strangely addicting and lots of fun.
Minecraft PE - Not as good as the PC version yet, but they're making strides.
Dead Space - Cool game, works great on the a100.
ShadowGun - Great graphics, solid gameplay.
Demolition Inc - Who doesn't like to destroy cities?
Bubble Birds - Free game I got from Amazon's FAOTD, that's a good time killer when you don't want to get too involved in a game. It's simple and stupid and yet I still find it to be fun.
Rooting:
Sorry, I haven't rooted mine so I can't answer this, but hopefully someone else can address it.
Enjoy your tab!