2 Weeks in Spain & Portugal (Itinerary Check)

depending on what you like, I personally would spend 2 full days in Lagos and surrondings and add a day to fantastic Sevilla.
You can do 3 days for Lisbonne but it's very tight as you will want to spend a day Sintra to visit a few castles and the old city of Sintra.
That's my very personal take on this.
I suspect you will get many different answers and they will all be good. :-)

I’m not sure if you have significantly accounted for connecting travel.

Day
1..Arrive Barcelona (4)
5..Fly to Seville (3)
7..Day to Cordoba for the Mezquita
8..Bus to Faro/Lagos then pick up car for the Algarve (3)
11..Drive to Lisbon (4) then drop off car
13..Day to Sintra
14..Day to Belem & Cascais
15..Depart Lisbon

Updated to show actual travel days and nights...

Wasn't planning on going to Cordoba or any other day trips. Probably just Sintra from Lisbon, if we feel like it. Do you think it's worth it to go to Cordoba from Seville and take away a day from Barcelona/Lisbon/Algarve?

Yes for cordoba, ans i would cut a day in Barcelona.
My take here is that by day 4 in barcelona, you will have done the major sites. All Gaudi related sites, el born district, old part etc, etc.
So if i have to choose between the Mezquita and jewish corner plus a few palaces in cordoba and the 4th day in Barcelona, i go cordoba

With 17 nights all up I would probably go 4-4-4-5.
Cordoba is definitely worth the effort.

Day
1..Arrive Barcelona (4)
2.3.4..Barcelona
5..Fly to Seville (4)
6.7..Seville
8..Day to Cordoba for the Mezquita
9..Bus to Faro/Lagos then pick up car for the Algarve (4)
10.11.12..Algarve
13..Drive to Lisbon (5) then drop off car
14.15..Lisbon
16..Day to Belem & Cascais
17..Day to Sintra
18..Depart Lisbon

Wow, thank you so much for your detailed reply!

I would never go below 4 nights in Barcelona. The arrival day for most of us is not good for sightseeing beyond the walking-around sort of thing, due to sleep-deprivation and jetlag. Four nights just gives you three real sightseeing days, and that is not a lot if you want to visit very many of the top sights. The need to buy most of those tickets in advance, thus being locked into fixed entry times, makes short visits to Barcelona quite challenging.

I would skip the Algarve entirely and go straight to Lisboa.
It is a fabulous city.

I thInk the above allocation gives you a nice pace everywhere. I agree with acraven, never go below 4 nights in Barcelona.