We use the form with the auxiliary to add emphasis to the statement, in order to make your point more forcefully. In the first example, you can imagine that the other person is angry with you for not getting in contact. You want to defend yourself. Maybe you feel that it's not your fault - perhaps you rang your friend's number but her phone was switched off. So you would say 'I did try to phone you' to emphasise the point that you made an effort - it wasn't as though you forgot to contact her. You want to make it very clear that you 'did try'. This is more forceful than the simple neutral statement 'I tried'.

I depends on how they ask the question or what question they ask. You can say: I tried to call you. But if they ask like they ar doubting you or think you did not call, then you can say: I did try to call you/

It's for emphasis, or contrast. "I did try to call you." (Don't think that I didn't try.) "I did find some information." (In spite of a long search.)