Sintra is only 30kms from Lisbon and is in fact now virtually a suburb of Lisbon.
I would suggest something along the following lines:
Full Day 1 -- Lisbon - See the Baixa, Alfama and possibly take the tram to Belem to see the Jeronimos Monastery and the tower of Belem.
Full Day 2 -- Lisbon - Take the Train from Rossio Station to Sintra (+- 60 minutes by train)- Full day in Sintra and/or take a bus to Cascais (Scotturb 403 every 75 minutes or so) or Estoril . Return from Cascais by train to Cais de Sodre station (+- 45 minutes). Wander up to the Bairro Alto for dinner.
Full Day 3 -- Lisbon - Tidy up what you didn't achieve on the first two days.
Full Day 4/5/6 -- Pick Up car at Lisbon Airport - Head North to Obidos, Batalha, Alcobaça and Nazare as public transport links to these from Lisbon are slow and convoluted. You might even have time to stay in Coimbra. There are IBIS Hotels in Leiria and Coimbra which might be good economic bases to stay at.
Full Day 7 - Head back via Santarem and the inland bank of the Tagus to Evora.
Full Day 8 - Day in Evora
Full Day 9 - Leave Evora and travel via Monsaraz (Fortified Hill top town) and Alqueva Dam to Serpa. (I recommend Casa da Serpa - see http://www.casadeserpa.com/ )
Full Day 9 - Travel via Beja and Mertola to Tavira.
Full Day 10 - Tavira and possibly old Town Faro or pop over the Guadiana to Ayamonte in Spain for Tapas.
Full Day 11 - Tavira
Full Day 12 - Lagos - Explore Lagos
Full Day 13 - Lagos - explore West Coast to Sagres.
Full Day 14 - Leave Lagos and make your way by car up the West Coast to Setubal. Stay at the IBIS just outside Setubal. Explore are and then head for Lisbon Airport (40 minutes).
I understand your concerns about driving on the wrong side. However, once you are clear of Lisbon, things are very easy as we have an excellent motorway system. To undertake the itinerary you want to do, hiring a car is the only realistic option if you don't want to spends hours grinding around on local buses or trains. This is particularly true when trying to visit Obidos, Nazare etc.
Check out some of my comments about this on my blog:
http://barlaventostravelblog.blogspot.com
http://barlaventostravelblog.blogspot.com/search?q=Obidos+Nazare
http://barlaventostravelblog.blogspot.com/search/label/Portugal%20Lisbon%20Praya%20de%20Rey%20Obidos
http://barlaventostravelblog.blogspot.com/search/label/Portugal%20Lisbon%20Nazar%C3%A9
http://barlaventostravelblog.blogspot.com/search/label/Portugal%20Lisbon%20Figueira%20da%20Foz
