Are the exercises necessary to understand the subject of a mathematical textbook?

If your goal is to become a research mathematician, then doing exercises is important. Of course, there will be the rare person who can skip exercises with no detriment to their development, but (and I speak from the experience of roughly twenty years of involvement in training for research mathematics) such people are genuinely rare.

The other kinds of exercises that you describe are also good, and you should do them too!

The point of doing set exercises is to practice using particular techniques, so that you can recognize how and when to use them when you are confronted with technical obstacles in your research.

In my own field, two books whose exercises I routinely recommend to my students are Hartshorne's Algebraic geometry text and Silverman's Elliptic curves text. The exercises at the end of Cassels and Frolich are also good.

Atiyah and MacDonald also is known for its exercises.

One possible approach (not recommended for everyone, though) is to postpone doing exercises if you find them too difficult (or too time-consuming, but this is usually equivalent to too difficult), but to return to them later when you feel that you understand the subject better. However, if upon return, you still can't fairly easily solve standard exercises on a topic you think you know well, you probably don't know the topic as well as you think you do.

If your goal is not to become a research mathematician, then understanding probably has a different meaning and purpose, and your question will then possibly have a different answer, which I am not the right person to give.

Depends on the textbook, I suppose. Some textbooks introduce a lot of material in the exercises that isn't developed in the main text.

I think that the most important point in mathematics is to think about the subject for long periods of time. If you think about mathematics, then you will often develop intuition which is very important. Of course, if you think about something for long periods of time, then your memory of the material is better as well.

Ultimately, the point is that people generally learn more by doing (compare active learning to passive learning). Of course, there are exceptions to every rule and you are the person who best understands your own strengths and weaknesses. The important point is to identify your weaknesses and work hard on them through a combination of active thinking and problem solving.