17 days in Italy first time

As you already have your train tickets Rome-Venice, its a little late to suggest that you fly into Venice and out of Rome, or if flying to Rome is 'fixed', that going directly to Venice and doing all your days in Rome at the end would have been better.

Is our itinerary doable. Are we crazy?

Yes. No

Just not the best arrangement/order of travel

We already bought airline tickets in and out of
Rome.

I understand now that we should have go to Venice directly but it is kind of late.

Any other suggestions?
Rome (August 16 to August 19) . 3 nights

Venice (August 19 to 22) 3 nights.

Cinque Terre (Vernazza) (Aug 22 to 25) 3 nights.

Pisa and go directly by train to Naples/Sorrento area.

Naples and Sorrento area (August 26 to Sept 1) 6 nights

Return to Rome on Sept 1(spend the night in Rome and flight back Sept 2

You have a lot of travel involved, but your plan is doable.

Personally, I would have spent more days in Rome and less in Naples/Sorrento and/or Cinque Terre. Further, you aren't visiting Florence???!!