In basic terms, ignoring certain situations like being significantly overweight or being completely untrained, you build muscle (and add some fat) during a caloric surplus, and you lose fat (and a bit of muscle) while being in a caloric deficit.

Naturally, it's logically impossible to simultaneously be in a caloric surplus and deficit. It's not possible to build a large amount of muscle while also being in a caloric deficit, and you won't ever lose a meaningful amount of fat without being in a deficit.

Building muscle requires an excess of calories, but unless your diet is supernaturally fine-tuned, you're gonna put on some fat at the same time.

Burning fat requires a deficit of calories, but you'll also lose a little bit of muscle mass along the way.

If you eat at maintenence, you won't have the excess nutrients needed to build new muscle, and you won't need to burn fat to survive either.

You can.

It is called body recomposition. A lot of people do it.

The problem is that while it is arguable more or less efficient than bulking or cutting, it is a psychological difference.

During a bulk & cut, you see massive differences on both the scale and your body in a short amount of time. That gives a ton of motivation and makes for good progress photos.

If you gain 5 kg mass, some fat, some lean in 6 months then lose 5kg mostly fat, some lean in 6 months for a total of 4kg lean muscle you will see much bigger differences than if you stay the same weight but trade 4kg of lean muscle for fat in 1 year.

Seeing the change is extremely motivating for most people and motivation and discipline are the two most important factors on fitness. They are what help you stick to your diet and training regime.

My general understanding of why you bulk and cut is because muscles take a lot of calories (carbs and protein) to grow and sustain growth. So in your bulk cycle you focus your workouts on pure muscle and strength growth and eat a lot which causes you to both have sustainable muscle growth as well as gain fat from the excess of calories you eat. Then in your cutting cycle you try to lose the weight you gained but when you are on a calorie deficit you also lose muscle mass. So, you focus your diet to hold as much muscle as you can while also focusing your exercises on high cardio. Again this is just my general understanding of why you do it I may have gotten some things wrong

I'm not a fitness guy. However, I'd assume it's for the same reason you don't press on the gas and the brake at the same time. I could be wrong, however.

You can, it's called a lean bulk.