Let's say I forgot something important and someone reminds me of it.
Do I say:
•How could I forget?!
or
•How can I forget?!

Are both correct? And do I have to put 'that' at the end? Like this:
•How could I forget that
or
•How did I forget that

See a translation

"How could I forget (that)" would be the most correct in that situation! You can use or leave off "that" it doesn't matter :)

Because you forgot something important in the past, you use the past tense to refer to it even if you're just now remembering you forgot it. Does that make sense?? haha

I'm having trouble trying to think of a way to explain "How can I forget", but I'll try!! Personally, I think that phrase is used in more of a joking, sometimes mean way?

for example, maybe someone you don't like asks if you remember them, so you say "Ooh, how can I forget you?"

Or if you genuinely don't remember someone but you don't want to seem rude -- "Haha, how can I forget you??!!"

BUT "How could I forget" you could also be used in those situations. "How can I..." just has a more specific context!

Hope that helps?!