1. You have to solve the English problems in your own way.
2. You have to solve the English problems on your own.
3. You have to solve the English problems for yourself.
4. You have to solve the English problems by yourself.
(Are they all grammatical? What is the difference among them?)
5. Who is taller, Tom or Sam?
5-1. Which is taller, Tom or Sam?
5-2. Which one is taller, Tom or Sam?
(Which one is correct?)
1-4 are all grammatical, yes. They all mean approximately the same thing.
Among the 5s, the first one is the best. 5-1 is using "which" to refer to people, which is not a good idea. 5-2 is OK, but 5 is better.