My mother spoke/has spoken to my teacher recently.
Both are OK- some grammars say we use the present perfect with 'recently', but we do use the past, though less so in the negative. If this is an exercise, I imagine they are looking for the present perfect as an answer.
Eddy gave/has given Melanie a present for her birthday.
It depends when the birthday is/was. If it was last week, then the past would be correct. If it's today, then the present perfect could be used.
Alan wrote/has written two postcards to his family,but they haven't arrived yet.
The past works better for me as the writing is finished- the problem is the post.
Has he written/Did he write a letter to his parents?
Again, both could be used depending on the situation. If he's sitting at the computer now, then the present perfect would make more sense to me.