In the present tense I would say "I forgot my house key" It's not technically correct but it's a real life usage. "I have lost my house key (and now I can't get into my house)." This example sounds very, very stiff and formal and too wordy to me.

Yes, However, it's in the past tense implying a completed action which might not have a result in the present. Example: I forgot my house key yesterday (and had to wait until my husband came home with his copy). If the loss of the key has a result in the present you could use the present perfect tense. I have lost my house key (and now I can't get into my house).

Yes, that looks okay.