Question

1. I forgot to lock the door.
1-1. I forgot that I would/must/should/had to lock the door.

(Are both the same?)

2. I forgot locking the door.
2-1.I forgot that I had locked the door.
(Are both the same?)

Answer 1

The first one is correct.

You could also say: "I forgot that I must lock the door." or "I forgot that I should lock the door." or "I forgot that I had to lock the door."

# 1 is the best.

Both of the sentences for # 2 are correct. They mean that you had locked the door and forgot that you had done that. You'd say this if you locked the door and forgot to take your key with you.