I think it would be very difficult to walk south on the Camino Portugues (I believe Margaret Meredith did this a few years ago and had some difficulty, but she is now on the Frances so I don't expect her to post on this for a while!). The crowds coming towards you will be very small, so you can't rely on seeing people to indicate which way to go.
There is no waymarked path from Faro that I know of. There is, however, a marked trail from Sagres to Santiago de Cacem (given the town's name and the fact that there is a Santiago in the local church suggests to me that it was at some point on a Camino). This route has recently been developed by the government -- some details here:
http://www.rotavicentina.com/?lang=en I was in Sagres last spring and saw markers and a walker or two.
As far as the weather goes, it's a crap shoot. I lived in Lisbon for two academic years -- 2004/05 and 08/09. The first winter was sunny, mild, glorious weather (bad for the land, the forest fires, and the crops but a delight for us). The second winter was payback. LOTS and lots of rain. You may get one of these two extremes or a nicer combination.
There are other posts on the forum about this, so if you do a few searches you will find a lot more more information. For instance:
http://www.caminodesantiago.me/comm...southern-coast-of-portugal.17932/#post-131232
http://www.caminodesantiago.me/comm...utes-starting-lagos-or-from.13689/#post-92467
http://www.caminodesantiago.me/community/threads/santiago-to-lisboa-nov-dec-2010.10111/#post-61956
http://www.caminodesantiago.me/comm...go-de-compostela-to-lisbon.20816/#post-163028