Does imperative, interrogative, modal verbs and auxiliary verbs affect the structure of the verb

[Ivancho]

I need to learn about a structure of a verb and ask If it has something to do with modal verbs, auxiliary verbs and verb tenses.

For instance, the verb carry on can be followed by
the preposition with and gerund, continue can be followed by an infinitive and gerund whereas keep and keep on can be followed only by gerund.

Is it okay to use them in an imperative sentences with modal verbs and future tenses?

For instance, imperative
1 Carry on with your homework!
2 Carry on doing your work!
3 Keep doing your homework!
4 Continue doing your homework!
5 Continue to do your homework!
Are all the sentences possible?

[Josef Essberger]

1-4 are all good. 5 is possible but less idiomatic.

[Ivancho]

Thank you very much! Can I use the same sentences as a request with can/could and in future tense with will/going to?
For example,

1 Can/could you carry on with your homework?
2 Can/could you carry on doing your work?
3 Can/could you keep doing your homework?
4 Can/could you continue doing your homework?
5 I will carry on with my homework.
6 I will carry on doing my work.
7 I will keep doing my homework.
8 I will continue doing my homework.
9 I am going to carry on with my homework.
10 I am going to carry on doing my work.
11 I am going to keep doing my homework.
12 I am going to continue doing my homework.

Are all these sentences okay?

[Josef Essberger]

These are all possible.

With 1-4 you have a question mark, meaning that they are all real questions. If, instead, they are polite instructions, the question mark is not normally used.