Is it possible to buy and sell coins, quickly and within those windows of time when you think its at a low or a high?
Or is it possible to convert between BTC and the other currency or asset within that timeframe on the exchange site or something like that?
Yes. That is what exchanges do. It sounds like you want to find one that offers as many trading pairs as possible. You should check out the altcoin sections and read about the different exchanges.
Also, what is the best method of buying BTC and then storing it?
The Trezor seems to be a popular hardware storage method. A printed private key in a lockbox isn't a bad idea either. Some people even keep their BTC on physical coins. Casascius coins being arguably the most popular.
What is the benefit of taking the money into a wallet, does it cost money to add/remove it from the exchange?
Websites have a bad track record of getting hacked at the end user's expense. Most will charge you a fee to withdraw funds, but it's worth it.