This is quickly going to become too complicated to tell you for sure if your deck will be able to support a structure like a pergola, but if it is even half the weight of your typical hot tub then your deck will very likely not support it.
To accurately determine this you need to figure out the additional #'s/sq ft of load that a structure as large as a pergola will place on top of the deck. Doing this is simple with something like a spa where you have a clear square footage area and a relatively even weight distribution when filled with water and people. This is not so easy to determine with a structure where certain points of the structure are going to carry the weight of the structure differently. The weight is not as evenly disbursed.
Right off the bat there I would advise you to consult an engineer with your structure plans to determine if this is feasible and safe. This is completely disregarding the fact that your deck is only as strong as its weakest point. If your house is brick and your ledger board is attached with carriage bolts then even if the joists were on 12" centers then you are still likely putting entirely too much stress on the ledger board. You also could have a lot of problems if any part of your structure is going to sit over that 2' overhang.
In the end, an experienced engineer will probably be able to confirm pretty easily if this will be feasible or not.
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?
Best Answer
A unloaded 19' long cedar 2x10 will sag about a little less than 0.2" which is L/>1000 and I would expect that to more than fit your needs.
HOWEVER: 19' long 2x10's will roll over in the middle meaning that they will twist in the middle with the appearance of the top tending to roll to one side as if attempting to roll completely over. The way to mitigate this is to add blocking. It is typical to add full depth blocking at 8' oc which would require two rows of blocking for your case.