Can "this, that, it" be referring to "infinitive or gerund" when it is the object of "do"?

For example,

1. I just thought it would be very hard for her to move the desk on her own, so I did that/it for her. ("That or it" is referring to "to move the desk")

2. To me, moving the desk seemed to be very hard for her, so I did that/it for her. ("That or it" is referring to "moving the desk")

3. Could you please help me to move the desk? I couldn't do this on my own ("this" is referring to "to move the desk")

If my thinking is correct, then, is it grammatically correct to say the last part of each sentence? as

A1: I did it to move the desk. ("It" is referring to "to move the desk")

A2: I did moving the desk. ("Moving the desk" is the object of "did")

A3: I couldn't do it to move the desk.("it" is referring to "to move the desk")