Learning: Novice investor. Learning about a lot, including options (call options). Questions about the basics of buying a call option.

So calls and puts are both types of options, that you can buy and sell.

I'm focusing on call options for the time being.

A call option is a contract to have the option, not obligation, to buy a stock at an agreed upon price (strike price) at any time during a window of time, for a premium, i.e. a price per share, and the motivation to do so is that the buyer believes the share price will not only increase but at least increase to the listed strike price before the expiration date of the contract?

Example:
BUTT is at $100 today. I'm going to buy ONE $110 call that expires in a month from today (Aug 10) for .50 (the premium) which is PER SHARE, specified at x 200 shares (is this “leverage”?). So that ONE call option contract will cost 0.50 x 200 = $100 for the premium, multiplied by the previously specified $110 strike price, resulting in a total “cost” of $11000?

As the buyer, I want the share price to go up, way up. So if, during the one month duration of the contract, the share price goes to $120, I can “exercise” my option to buy at the agreed upon strike price of $110 at the total cost of $22000, although the overall intent is to turn around and sell the 200 shares at the current share price of $120 for $24000 and netting an overall profit of $2000 after having risked $100?

My maximum loss is the premium I paid in the event the contract expires worthless when the share price hasn't gone up enough?

My maximum gain is that from however many calls I purchased? So if I'd bought 5 calls under the same conditions I'd have risked $500 for $10000 profit?

Obviously hypothetical numbers and they're far different IRL but conceptually I wish to understand this.