For newbies such as yourself, generally the best trade off between security and ease of use is a reputable hardware wallet such as Ledger or Trezor. Unfortunately you will need to purchase one of these devices for around $50, but if you wait for Black Friday in a few weeks they have previously had some very good deals on (up to 50% off).

This is a fairly standard way of buying and then holding bitcoin. You don't have to do it right away, and some people might prefer to build up a little bit of bitcoin on their exchange account over time before withdrawing it all at once to save on the exchange's withdrawal fees. But the less time your bitcoin is held on the exchange, the better.

No. You will still be able to see your purchase history and withdrawal history, but it will not show your balance. Your wallet will show your balance instead.