We fly into Porto. We will likely fly out of Lisbon.

We are set to spend 7 nights in Lisbon which can be lengthened but not shortened.

Original plan was 12 nights total. We were going to spend 5 nights in Porto but it now seems we will have a few more nights to play around with while planning.
[We are not yet locked in to 13 nights; we could add on 1 or 2 more.]

How would 2 (or 3) nights in Porto; 2 nights in Coimbra and 2 nights in Óbidos taking trains between each and then 7 nights in Lisbon?

What would you suggest as an itinerary? We aren't planning on renting a car, but could if it makes us able to see things that are otherwise not visitable.

Architecture, history, food, natural beauty, arts/culture are all of interest; shopping not so much.

Thanks in advance and extra points for places to stay in Coimbra and Óbidos.

Obidos is worth a couple of hours tops.

I can help you with the north part of Portugal

Near Porto you can easily take the train and visit Braga and Guimarães.
Allow 1-full day to visit the Douro Valley (book a tour if you don't plan to rent a car).
On your way to Lisbon, I believe it is better to spend one night in Aveiro and 1-2 nights in Coimbra.
Obidos is far away from the train railway and there is not much to visit besides the castle (1-2 hours max).

Porto, Braga, Guimarães, Aveiro and Coimbra all have plenty of Architecture, history, food and arts/culture
Douro Valley is history, natural beauty shaped by men (vineyards), the river (Douro) and food.

If you decide to stay extra days in Portugal, you could also include Minho (Ponte do Lima e Viana do Castelo) and Gerês the only portuguese national park.

One of the highlights of our trip to Portugal a few years ago was Evora, about 2 hours east of Lisbon, in the beautiful Alentejo region. You will need a car. Evora is a beautiful walled city surrounded by a region of cork trees, and also in that area there are megaliths - ancient standing stones like Stonehenge. They are not marked and there's no information at the sites but there is a fabulous tour with a guy named Mario. We've recommended it to 3 or 4 friends and every one of them came back raving about it. http://www.eboramegalithica.com
We also stayed at a lovely small hotel with the best breakfast I think I've ever had. It was 3 years ago and we still talk about it. Albergaria do Calvario. http://adcevora.com. We were only there for 1 night and it was not enough (although better than nothing!), but 2 would probably be fine.

You do not have to have a car for Evora. Take the bus (rather than the train) from Lisbon. The Albergaria can arrange car tours for you - both the megaliths and wine tasting. I am sure other hotels can do the same. Do not go just for the Roman temple - very disappointing, Conimbriga outside Coimbra is much more interesting.