Question

1. I remember writing a letter.
2. I remember that I wrote a letter.

3. I remember to write a letter.
4. I remember that I have to write a letter.

Does #2 mean #1?
Does #3 mean #4?

In #4, can we use 'must' or 'should' instead of 'have to'?

Answer 1

#s 1 and 2 mean the same thing, yes.

#3 would be better if you make the verb future: "I will remember to write a letter."

#4 is fine, and you can also substitute "must" or "should."