있다 - To exist, to have. This is the stem form of the verb. same as 공부하다, 먹다, 마시다.
있어요 - Is the present form of the verb (polite- to all levels).
있었어요 - Is the past tense form. So its like saying "it existed, I had it"
있었었어요 - Is the past form, however it means "it existed, but is gone" or "i had it, once" It is to indicate that something that happened in the past has occurred and ended.
so in this logic:
공부해요 (하다 changes to 해요) when conjugating.
공부했어요 - I studied
공부했었어요 - I studied, then I didn't (sic). But it would be interpreted as "I studied (a long time ago, and haven't studied again).
Also, if i ask "카카오톡 할래요?" "나랑 메일 할래요?" Is it correct?
I'm not a native speaker, but it seems right. Your asking "would it be okay to email me?"? is this right? I think you could also say:
"나는 메일을 할 수 있어요?" "can you email me?" But i'm not 100% sure about that. This is how I would speak it, but i would leave out the 을. Hope this helps.