Travel Advice

Sorry, but you will be spending longer in transit than actually seeing places, as you are trying to cover far too much ground. Mid October weather can be mild, but I have also had torrential rain near Lagos too. I would drop the Algarve and focus on the area between Porto and Lisbon. Add one of the Lagos nights to Porto, particularly if you are likely to be arriving with jet lag and add the other 2 nights to Lisbon.

Evora can be covered from Lisbon as a day trip.

I'm not sure about Lagos (haven't been there), but the rest are easily connected by train.
You can always visit Evora as a day trip from Lisbon if you want to reduce hotel changes (easy by bus and train).

I did read somewhere that I should skip Algarve. But I am a beach bum and would really love visiting that area and explore caves on kayak, however I am not sure how forgiving the weather would be to do that in October.

As to whether your plan "sounds reasonable": I hate to be blunt, but I'm afraid it doesn't! Two nights in Lisbon and Porto (and I do commend you for having timed your plan by nights, rather than by days -- it makes it much easier to evaluate) are way too skimpy for these great cities, for two nights mean only one full (non-travel) day.

I'm a bigger fan of Coimbra than most people, and gladly spent four nights there myself, but I realize that most tourists won't favor such an extended stay, and one night may give you time at least to see the old University, the city's main tourist site.
I didn't visit the Algarve on my one trip (so far) to Portugal, so I can't recommend anything in particular, but given your preference for beaches, you could probably spend your entire nine-night trip exploring that one region, plus Lisbon.
(I'm not hip to the current craze for acronym-speak, and so have no idea what "CP" is, or how it might affect your itinerary, though I would guess that the P stands for "Português.")

CP is the train
https://www.cp.pt/passageiros/en

Faedus has a good suggestion regarding perhaps focusing more on the Lisbon area and South instead of trying to cover North and South.

And, if you happen to like big waves, you might get lucky with seeing them at Nazare.

If you want beach time, look at the Silver Coast.

October isn’t the time to be visiting the caves in a canoe. The currents are very strong and I saw what was nearly a fatal incident one year in June when 2 fit looking 20 something old men had to be rescued as they were exhausted. The sea is cold as it’s the Atlantic not the Med.

Thank you everybody. I was trying to follow this itinerary -> https://www.ricksteves.com/europe/portugal/itinerary. I will work with my partner and revisit our plan.