How many increase

Increase is not a countable noun, so you can't say how many increase.

"How many of the increased number of jobs go to Americans?" (How many refers to jobs)

Neither are correct. You need to use:

What increase in the number of jobs goes to America?

Or (depending on your context/meaning);

What increase in the number of jobs is attributable to America?

“An increase“ is a single concept, with a particular value, and you cannot refer to “many increase” as if it was a plural. Notice that the question focuses on “the increase” (singular) not on “the jobs” (plural).