Am i understanding options correctly?

So i've researched how option trading works, but this one part is a little fuzzy and i can't find a clear explanation anywhere.

Lets say stock “a” is price at $100 on 1/01.

I buy a call option that expires on 1/15 with a a target price of $110.

lets say on 1/12 the price of stock “a' goes to $120.

This would be in-the-money, now my question is, am i able to exercise early and still reap the rewards? because i think this stock might go back down below $110 by 1/15.

Additionally, if i exercise early on 1/12 when the stock is at $120, would i recieve less money if i exercised on 1/15 if it remained at 120. Essentially does the “expiry date” act more like a “if the stock reaches your target price before this date its good”, or “if the stock reaches your target price ON this date your good”..

Sorry if this a little hard to understand and jumbled, but if anyone could answer this, that would be very helpful. Also if you want me to clarify anything just comment. Thank You.