1. In judging whether a ring is aromatic or not, I like to look at whether all the atoms around the ring are sp2 hybridized or not, not whether you can form a system of alternating double bonds around the ring.
2. You are correct that the nitrogens would normally be sp3 hybridized, but atoms with lone pairs can change hybridization from sp3 to sp2 by moving the lone pair from an sp3 orbital to a p-orbital. Although this costs energy, the energy gained from aromatic stabilization outweighs the cost of moving the lone pair to the p-orbital.
So, yes, caffeine is aromatic.