I visited Portugal about 8 years ago with my BFF (the year we turned 30). We spent about 12 days there, so a similar time period. I'll describe my itinerary, and then make a suggestion.
We flew into Lisbon, rented a car, and left immediately for the walled city of Obidos. Obidos is about an hour or so north of LIsbon - it's very "touristy" but it's a low-key place with some pretty churches, castles, etc. to look at. We spent 2 nights there, and then left for Douro Valley for 1 night. We then drove to Bussaco Forest for 2 nights, a stop in Evora for the day, then to Marvao for a night, then to the Algarve for 3 nights. We spent the last 4 nights in Lisbon, dropping the car as soon as we hit the airport.
We did a day trip to Coimbra, but I can't remember if it was en route between Obidos and Douro Valley or otherwise. I remember walking around a university and doing a little pottery shopping, but I don't think that we really spent a lot of time in Coimbra or found it to be overwhelmingly interesting. We also went to Nazare, Batalha, and Fatima.
We did a lot of moving on our trip, but the car made it easier/less disruptive - and of course, I didn't have small children with me :-). We also stayed in pousadas, the state-owned old castles or stately homes -- it was beautiful and historic.
In Lisbon, we stayed in the Alfama area, and liked it very much. It was well-located. Baixa is "in the middle of everything" so that may work best for you. We stayed at Albergaria Senhora do Monte, which was a hotel. It had great views, was clean, and had a rooftop bar that was awesome (breakfast was served there, and it was also open late). I absolutely loved Lisbon, and I'm jealous that you're going! I felt like I could have spent more time in Lisbon itself.
I wish that we would have gone to Porto.
- Why are you interested in going to Evora? It's quite out of the way for a 1 night trip given the other locations that you are visiting. If I recall correctly, it's a few hours from Lisbon - like 2 or 3? The highways are terrific, although I know that you mentioned taking public transport, which I have no experience with aside from the subway in Lisbon. The chapel of the bones was interesting, and the scenery (cork trees) was very pretty as we entered the area, but I can't say that I would have considered it a "don't miss."
- Could you consider grouping your "Lisbon" stay and Sintra side trip at the back-end of the trip -- it'll reduce the number of times that you're changing hotels for the night? Maybe land in Lisbon and take off immediately for Coimbra, Obidos, or Nazare? I've found that I normally don't mind a brief train ride right after an international flight.
- If I became the master of your trip, I would skip Evora, and concentrate on the northern section, surrounding Lisbon, Obidos, Coimbra, Nazare, Porto. I also really loved the Algarve (Salema) although you'd end up with the same situation with it being quite a distance from where you're going and I don't know that the weather would be ideal in February.
Good luck -- hope that my comments were helpful.