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trees and plants

I try to move away from it but i need help from someone.I read things from a book now that is about algebra, geometry, topics on convex sets, plane geometric figures, hermitian harmonic functions, quadratic forms, embeddings, connections,curvatures and geodesics on riemannian manifolds. Some people focus only on a very little interest on math or physics mostly up to high school math and physics from popular science books.

What i want to ask is: what should be the correct view i should have on math and physics? If i study about other fields in math or physics like representation theory, Lie groups,category theory , proof theory which contain results on math but other people may find uninteresting will it be worth it? I think it will, but i am not sure for some reasons. Also, if i study math only for math not for their applications to fields outside of math, or physics only for physics and not for applications outside of physics like engineering, technology, industrial uses, is it ok?

Also, some people talk about theorems in math and proofs in physics but if someone reads the textbooks then also examples, applications are provided, which are needed. They could be considered theorems but they are called examples or applications. Other question: personally i think as long as questions are made in math and physics , they will not reach an end and people will not stop doing research on them, but some popular opinions say other things i think. What is correct?

In my high school and junior high school as i remember importance was not given so much on proofs in math and physics but on the statement of the theorem not the proof so much. But proofs are very important in math and physics and focus on them is worth it as i think.Should focus on teaching be more on proofs than it is until today?

Some things in math and physics are very popular like higher dimensional riemannian geometry, differential equations, general relativity, quantum mechanics they are ok, but some people i think know or understand a few basic things on them and do not study them more. Others consider other fields of math or physics or topics not so much worthy. I am a confused from these things. Can you help me solve this confusion?

And about the importance or interest of a question and finding its correct answer in math or physics, what are the criteria for the result to be accepted?Thank you.