Do US universities offer Graduate Paid Internship programs in CS/CSE?
Yes, US Universities do offer Graduate paid internship programs in CS/CSE.
It is called Curricular Practical Training/Optional Practical Training.
But each university is different in accepting your CPT/OPT.
You need to have a job offer in hand, check with the DSO of the university for the dates within which you need to apply for your CPT/OPT.
You need to be atleast 18 years i guess.

For more information you can look this up : http://www.ice.gov/sevis/practical-training/