I don't know about the ports used by the bitcoin client but it should work if you can download the blockchain and see a number next to "connections".
You also have to wait a bit for the transaction to show and maybe double check your receiving address.
And yes you can use your wallet where ever you want. Just take it with you.
BUT be safe while you copy around your walled.dat file. Do not leave it lying around and do not lose it on the way
PS: Read this article about working with your wallet (security etc.): https://en.bitcoin.it/wiki/Securing_your_wallet
There you can also learn where to locate your wallet:
The data directory is the location where Bitcoin's data files are stored, including the wallet data file.
Go to Start -> Run (or press WinKey+R) and run this:
BitCoin's data folder will open. For most users, this is the following locations:
C:\Documents and Settings\YourUserName\Application data\BitCoin (XP)
C:\Users\YourUserName\Appdata\Roaming\BitCoin (Vista and 7)
"AppData" and "Application data" are hidden by default.