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?
You could probably get away with two 4x4 posts on this, since the house would pick up a good amount of the load. I would run 2x6 trusses and cleat them to your house (someone will eventually want to hang something off it, even if that something isn't crazy heavy, so don't toenail this end).
I would laminate a 2x6 beam to go between your 4x4s. The problem you face here is that 25ft is too long for a single board to really span well. Ideally, I would want this beam to sit on top of the 4x4s, since it will be heavy as well. To make a laminate beam, buy the longest 2x6s you can carry (12ft sections should ensure you're not needing to do a lot of bolting). You'll want some lag bolts and construction adhesive that can work with pressure treat (even if you use cedar, the PT stuff is coated for outdoor use). Stagger the boards to make one 25ft length, gluing and bolting as you go along. You can then put the beam on top of the 4x4s and use screw plates to attach. You can then toenail the truss boards (with deck screws) to the top of the beam
Best Answer
For a beam that long deflection will control over strength. A 2x6 HF#2 would pass code with 0.6" sag in the middle if you are ok with that. Going with a 2x4 will result in a deflection of about 1-3/4" and it will want to roll over in the middle since there are no bracing members. If you go with cedar or pressure treated, it will be more.