Where did https://launchpad.net/~bitcoin/+archive/bitcoin just install to? I just ran the following three commands:

sudo add-apt-repository ppa:bitcoin/bitcoin
sudo apt-get update
sudo su bitcoinuser
sudo apt-get install bitcoind


and I believe bitcoind has been installed on my Ubuntu 12.10 64Bit ec2 box, but the problem is I can't find it anywhere.

Where does bitcoin download to when installed via the ppa?
Just as all applications you install using Aptitude (apt-get), it is installed somewhere in a /usr subdirectory. However, where exactly doesn't matter because you can run those programs just by typing them in the command line.

Thus, you can launch them by typing in your Terminal either bitcoind for the server daemon or bitcoin-qt for the GUI Wallet. Also, most desktop manager automatically place a shortcut for all apps in their app menu; for Ubuntu Unity, you can just type "bitcoin" in your Dash to find Bitcoin-Qt GUI wallet app.

Just as most apps, it makes a folder in your home directory to save the data it uses. Both bitcoind and bitcoin-qt save all the block chain data in a ~/.bitcoin/ folder. You wallet is thus saved in ~/.bitcoin/wallet.dat.