Hi, glad to know that you've been reading up on the different types of wallets that exist. It's important to do your research before hand -- otherwise you could end up with big problems. Here are the answers to your questions:

1 - Yes. You can transfer any amount of Bitcoin from one place to another. This is because Bitcoins, no matter which location they are (laptop, web, etc.) are all stored on a Bitcoin address (and its private key). In order to transfer this, just open your wallet on your computer and go to the send section. Then, from whatever destination you'd like, such as web, android, paper, etc. make sure you take one of the addresses that it generates for you and input it on the send section. Then press send. Your coins will have been transferred.

2 - No. You need internet access in order to signify to the Blockchain that you would like a transfer of Bitcoins. Then a miner will process the transaction for you. This is for all wallets, but evidently you won't even be able to access a web wallet without internet access.

3 - It depends. Again, it doesn't matter what is "under your name". Bitcoin is totally anonymous, so all that matters is your wallet address. You can choose whether you wish to attach a transfer fee to the transactions. If you don't assign a transaction fee, it is possible that the transaction will never take place and get stuck in an infinite loop - so be careful. The higher your fee, the faster it gets sent. A fee is only charged ONCE per transaction, BTW.

4 - There are many reliable exchanges out there. For starters, again, Bitcoin is super anonymous. None, that I am aware of, require any personal information. Obviously, Blockchain is completely secure and just using Bitcoin requires no information. If you want to use an online wallet, you can use Xapo or maybe Coinbase - both are secure and don't require info to send and receive. They do require info to deposit but that's obvious because you need a CC or bank account to deposit. An exchange to sell Bitcoin for altcoins is Poloniex. This is my favourite and requires only an email, password and BTC address.