So, the storyline is that while you are learning the ways of the world, the village is dying. My question is, is there actually a definite deadline by which you have to return the GECK to the village, or the village will die and you will fail?
I'm not certain. I don't think so though. FO1 had such a deadline, but it was clearly stated. I'd think that if there was an actual deadline, it'd be clearly stated somewhere. RPGs tend to be like that: Time has little or no meaning in terms of quests. There are occasional exceptions though.
Or can you basically just play the game at your own pace, following up on as many random encounters and side quests as you want to, along your path of looking for the GECK?
That's pretty much what I did.
In regards to other questions, how useful is the 'science' skill?
Science is good for fiddling with electronic equipment. I don't know if it's useful enough to be a Tag skill though. If you're going for more of a "theme" character than an "uber" one, there's no harm in tagging Science.
My first char was brains rather than brawn, and had tag skills of Melee (reluctantly; originally it was to be Outdoorsman), Doctor, and Science.
Melee is *difficult*, IMO.
Doctor is a LOT better than First Aid.
The same question for 'Doctor?' It's more useful in the early going than Science, but it's effects can be mostly duplicated by other means, so maybe it's not worth being a tag skill, either.
I used Doctor to heal up as much as possible between rests. e.g. Use doctor until exhausted, then sleep to regain hps. Rinse, and repeat until healed.
Doctor is also much, much cheaper than healing objects - which are somewhat limited.
Opinions? [My second char, btw, has Small Arms, Melee Weapon, and Steal as Tag skills, with Gifted and Good Natured as Traits]
Gifted is one of my favorite Traits. Extra S.P.E.C.I.A.L. points are always a good thing. 
BTW, does this forum have a 'Search' function? I haven't found it yet, if so.
No. IIRC, "search" is only available if you have a "Premium" account and pay N54 money.
[o: *LEMMING* :o]