Best way to get to Rock of Gibraltar

I agree with the previous poster - you are really cramming a lot into a short amount of time. If you must see Gibraltar, you drive to a parking lot across the airport from Gibraltar. You then walk across the landing strip. It is rather time consuming and, I wish we had not spent the entire day doing it. It will also be a pretty long day driving from Seville.

I would also suggest rethinking Barcelona. I know it's a 'must see', but we ended up dropping it on our first trip to Spain because it is far from Madrid and Andalusia. I did go back a couple of years later to see Barcelona. Nazare in Portugal is not all that wonderful. And Tavira is very far from the other places you mention in Portugal.
I would encourage you to think about your top 2-3 priorities in each country. And remember you can do day trips from Madrid, Sevilla and Lisbon. I have spent more than 15 days in each country on trips and still felt I didn't have enough time!

Appreciate your feedback .. will revisit my itinerary

If you want to make this an enjoyable trip rather than an endurance test, you need to cut about half your destinations, or allow 3-4 weeks for the whole trip.

One way to start is to pick Spain or Portugal rather than trying to see places in both. As you know, connections between the two countries are spotty. Yes, you can fly between them, but you lose much of a day when you account for getting to and from the airports, waiting time, etc. However, connections within each country are good.

As you've also discovered, Gibraltar is far from most other destinations in Spain, and takes a few hours each way to get to. And you can't even fly there from within Spain!

If you pick Spain, look into flying into Barcelona and traveling through Madrid to Seville/Granada, or vice versa. You will have to change planes, but will save time compared to flying in and out of Madrid.