Edit: not safe anymore, read
https://bitcointalksearch.org/topic/walletgeneratordotnet-scam-paper-wallet-generation-scam-5461181Yes, they are legit.Last question, if I download the generator from GitHub, and launch it without internet - can I move coins to it safely, and expect them to be on the paper wallet?
That's actually the best way of generating your paper wallet and avoiding any possible malwares. So, yes. Your generated address will work.
That I won't be sending them to an invalid or wrong address, that the paper wallet generated is legit, and I will indeed - have my coins, in my control.
No worries. You will be safe. Download the source code and generate a wallet while offline if you want to be 100% safe. You can also generate a temporary paper wallet, send a small amount of BTC and test the private key to see if you are really in control of your coins. Then, just generate a new one that you will keep it safe.