1 week in Portugal

Your idea of flying into Lisbon and out of Porto is a good one with your limited time. But, due to the limited amount of time, you are breaking up your trip more than necessary and spending too little time in Lisbon. Your one night stays in Sintra and Coimbra could be eliminated easily. Sintra is an easy day trip from Lisbon and with a day (or two) extra in Lisbon you would possibly be able to pick a clear day for Sintra. If you spend the night you are stuck with that day despite the weather.
Sintra is much, much more enjoyable on a clear day and the biggest problem with April is the increased likelihood of rain.
You could also eliminate renting a car by staying in just Lisbon and Porto. There are day tours into the Douro that are quite nice and there are a lot of wines other than port in that region. You could also take the train into the Douro valley and possibly fit in a hike.

And, if you stay as many as 3 nights you could stay in a vacation rental apartment instead of hotels - which is my preference.
Obviously, this is just my opinion, but it might give you food for thought

We've recently returned from the RS Portugal tour. We loved Portugal-the sights, people, food. You don't have a lot of time and there is much to see, so you'll need to go back. We will! Lisbon has a lot to see. Sintra is a great day tour, but I would not spend a night. Logistically, you can do it yourself but getting there early is essential. Another option is a guided tour if your budget allows. We used http://www.yourfriendinlisbon.com and it was terrific. Lisbon is very hilly-you will get a good workout walking and there is lots of that. Do visit https://gulbenkian.pt/museu/en/. A great museum, surprisingly uncrowded, convenient to the metro. Another must see is Saint Jeronimos where you can see the tomb of Vasco da Gama (remember from junior high?). We also splurged on a culinary walking tour with Culinary Backstreets, seeing neighborhoods we would not otherwise have seen, eating strange and wonderful foods, meeting great people, but it is costly. I would heavily weight your time in Lisbon, then go to Porto. Don't close the door on Port until you have tried white port, aged tawny ports. The wines of Portugal are good and very inexpensive. As the previous poster said, there are day trips to the Douro Valley. We did stay a night here in the Douro Valley: http://www.csvintagehouse.com. Not a typical RS tour hotel, believe me. It was fabulous! Coimbra was alright, but, in your limited time, unless there is something you must see, I would pass. Do get Rick's Portugal guide. It is new and has very good ideas including detailed walks in Lisbon and Porto. I had other guidebooks as well, they were worthless by and large. For background reading I found this book terrific: The Portuguese by Barry Hatton. I'm re-reading it now that I'm home and getting even more out of it. Learn how to say hello, goodbye, and thank you and you are set. English widely spoken and they all know Portuguese is impossible! Have fun and feel free to PM me if you want other details.