Don't use bitcoin qt then. I agree it is a bit of a ballache for new users but I don't really think there's any need to use it with all the lite clients we have nowadays. Get one of them or a blockchain.info wallet and you're good to go immediately.
No, I will never leave my coins and private keys on a website I don't control. Maybe that's fine for people who use small amounts of Bitcoin, I use hundreds of thousands of dollars worth of Bitcoin so it simply isn't an option.
I'm going to have to go ahead and advise against storing hundreds of thousands of dollars worth of
BTC on a laptop connected to the internet.
It doesnt have to be. He can move the wallet.dat away from PC and to a secure drive while he's not using it. Right now it's important for OP to sync with the network,
nothing more. Later he can just return the wallet.dat to its place.
At this point i would delete everything but the wallet itself, just copy it and return there later (check manualy for roaming/bitcoin/), and reinstall core. There's no chance router is giving you problems with port now if it worked before..
Check out Electrum if you're considering to change you client, im starting to move there also, but mainly because core is slow, not because i think its bad overall.
cheers