I've got a problem with contacs:
-Does the contact always knows a consultant or specialist every where you go, even in a different country?
Yes.
The specialist is a problem for me: He chooses a Class ability, that I understand. But what if the class ability requires a skill check to use, or just provides a bonus?
Example I: I want to make a Specialist Scientist with the "Research project Elbow Grease class ability. How do I use such a specialist when my team is on a mission?
Pretty much this can be done as the definition of Phone A Friend.
Example II: I want to create a Specialist Hacker with the L337 class ability. How do I use such a specialist when my team is on a mission?
You contact your contact prior to the scene you think you're going to need to use the ability.