I am not perfect in English but I think you have to use "to" when you explain your reason to talk (in your exemple) with a verb (improve). If you had explain with a word like "pleasure" for instance, then you had to use "for" before it. Do you see what I mean? so, this is: " I would like to talk with foreigners to improve my English" " I would like to talk with foreigners for (my) pleasure" to me, the sentence " I would like to talk with foreigners for improving my English" is incorrect. Maybe you should wait for confirmations.

The difference, I believe, is that the subordinate clause came after a verb. I run to improve my health. I drink tea to calm myself. The pattern is <verb> <to> <verb in subordinate clause> If you the clause comes after a noun, you can use "for" I use a mop for cleaning [Also correct - I use a mop to clean] I use a medium point pen for writing letters. [Also correct - I use a medium point pen to write letters.] I can't assure you there is a hard, fast rule for this, but "to" definitely sounds better in this case.

Am I not right? does this form exist too? I have never seen that before, I have learned something...

There's no difference as i know.