I left out the following doubts. I really hope you can consider them, too.

1) He has grown out of his shoes, so we'll need to buy him new ones.
Hold the line is the same as hold on the line?
2) It wasn't necessary you bought the milk (wrong?).
You needn't have bought the milk (correct): there was still some OR there was some already (both correct?)
3) He would like to ask her out next Saturday (not the following Saturday?).
4) It is possible that he will be invited to dinner: he may (not "could/can") be invited..
5) I'm unsure if I will pass ... I might pass .... (NOT:I may not pass it).
6) He helped me zip up (is it possible?) OR He helped me do the zip up.
7) I'm trying to figure out a way of paying him for/back/all (which are possible??) the money I owe him.
I moved into a new house (is home possible?)
I packed my suitcases/luggage and I moved out (of my old house??)

1) He has grown out of his shoes, so we'll need to buy him new ones. OK

Hold the line is the same as hold on the line? No, sorry, they're not the same.

2) It wasn't necessary you bought the milk (wrong?). Yes, this is incorrect.

You needn't have bought the milk (correct): there was still some OR there was some already (both correct?)
Yes, these are correct.

3) He would like to ask her out next Saturday. OK

4) It is possible that he will be invited to dinner: he may be invited..
OK

5) I'm unsure if I will pass ... I might pass .... (NOT:I may not pass it). All three expressions here are correct.

6) He helped me zip my jacket.

7) I'm trying to figure out a way to pay him back all the money I owe him.

I moved into a new house (is home possible?) Yes, "house" or "home" are both possible.

I packed my suitcases/luggage and I moved out (of my old house??)
OK, with or without the wording in parentheses