Learn English – Explanation needed: Why is “and” wrong in this sentence

"Light", as an adjective, can certainly mean having an abundance of light, i.e. not dark.

In your context, it is perfectly clear that you mean that turning on the lamp makes the room lighter. To that effect, the following is fine:

Having more windows, the kitchen was lighter than the living room.

Using the word "brighter" is more obvious without context because it doesn't carry the ambiguity of "lighter", that being that it can also mean "less heavy". Everyone knows that turning on a lamp doesn't reduce the weight of the room so, in your example, lighter works fine.

In summary, both are fine but if you think context is unclear then "brighter" would be more obvious.

You are right. Since is a possible answer here, meaning because, just as you suggest.

However, if the instruction or task is to choose the "best" answer, then as is "better".

The sentence uses the phrase more and more, which suggests a process of increasing over time. As can mean while so is more likely in the sentence.

If we began with Since, meaning because, we'd more likely leave out more and more or say something like Since he exercised more and more, his muscles became more and more developed.

Note: In any case, learning facts like "differences between as and since is a very inefficient way to improve English proficiency. All possible such rules in English are endless in number and complexity. Reading more things you like at a level that's easy for you is much better.