忘れた。忘れてた。の適切な英語を教えて下さい。
This is very confusing to me.

case1 忘れた
A: Do you remember the name of the first speaker?
B: No. I forget/ forgot/ have forgotten/ don't remember.
(which is the most natural?)

case2 忘れた\忘れてた
A: Have you brought your umbrella?
B: No. I forget/ forgot/ have forgotten/.

case3 忘れてた
A: Oh, today was your birthday!
Sorry I forget/ forgot/ forgotten/ did't remember.

case 1:I forgot and I don't remember is natural case 2:I forgot case 3:I forgot

I forgot is usually most common.

I'd like to add that in case 1, I would simply answer "no" or "no, I don't". forget and remember are redundant in this case.

you can also say "i always forget"
example:
"did you bring your books this time?"
"no, i always forget" or "no, i forgot (again)

We use forgot in all those situations because the thing was forgotten in the past. The action is already completed. The memory has already been lost from your mind.
When you are actively trying to remember, but you can't, you can say "I don't remember."

Examples:
Where did you put your keys?
I don't remember.

How did the movie end?
Sorry. It's been awhile. I forgot.

Did you bring an umbrella?
No I forgot/left it at home.

What was the name of that guy you were talking to yesterday?
The guy at the bar? Sorry, I don't remember/I forgot his name.

what does いけね mean?? just curious :P