Adding a pergola to an existing deck?
First answer! (On an old question.)
Could the side beams that run perpendicular to the house extend almost all the way to the house? That would provide an additional 3.5-4’ of coverage along the house side, where we typically place deck furniture.
I'm assuming you mean the rafters (that rest on the beam) would cantilever over the 17' beam by 4'? That will be fine, as there won't be any weight bearing on the rafters other than their own weight.

Are 4x4 posts sufficient for the pergola?
In my estimation, yes. Check out load tables to be sure (capacity varies by species, length, wet or dry conditions, etc.) They will look a bit spindly though.

Is the placement of those posts OK to carry/support the load to the structure below?
Should be fine, in fact, you could place the posts almost anywhere. Each post will be carrying less weight than the average American male.

Any other suggestions or questions I should consider?
Pergolas don't weigh that much, but they do catch a lot of wind so lateral load and rotation of the posts is a real concern. I understand you're bolting some the posts to the joists so that should solve that problem. 

The 17' beam span will likely cause the 2x6 beams to sag. I'd recommend bumping up to 2x8 or 2x10.

By this point, you've probably built your pergola. How'd it turn out?