How do call and put options really work?

Let's say I buy 100 shares of XYZ at 7 dollars per share. Then I sell a call for a 6 dollar strike price. Can the person who bought my call exercise it right then and there, or do they have to wait until the price actually hits 6? If the price falls below 6 and then comes up above 6, and the contract is not executed,, did my buyer miss an opportunity? It makes sense that the person who bought my call would wait until the price went way up so he/she could buy my shares for 6 and immediately sell them for 7, 8, 9, or whatever quick profit can be got. Then again are there limits, rules, protection of some sort?

Same with puts. XYZ sells for 7, I sell a put for 6. The contract is not executed until the price drops to 4. Is this part of the risk, or is there some kind of limit?

The call buyer can exercise anytime they want for most underlyings. The reason they USUALLY will not exercise early will be because of the extrinsic value. Means if you sold the 1 dollar in the money call for 1.50 it still has .50 left of extrinsic value. If they exercise the option they will give up that extra value....and you will have your stock called away that you bought for 7 and get to keep ALL of the 1.50 credit received for net gain of .50. If the exercise the call they take possession at 6$ and lose the 1.50 value of the call for a net loss of .50.