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 those are possible. Converting BTC means trading it with other coins and that includes trade fees.
Also, what is the best method of buying BTC and then storing it?
If you live in the US, there are a lot of ways to buy bitcoin. You can use Coinbase, Localbitcoins, Coinmama, GDAX and more.
What is the benefit of taking the money into a wallet, does it cost money to add/remove it from the exchange?
There are some exchanges who has deposit and withdrawal fees and there are some who offer it for free. However, you still have to pay transaction fees in the Bitcoin blockchain. Taking your funds into a wallet may not be necessary if the exchange offers a direct to fiat withdrawal
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