You mean F(t_1, t_2) where t_1 is the decision time and t_2 is the time of the event whose utility is weighed. Yes, that's the general form, but we assume that discounting is roughly constant across time (ie depends only on t_2 - t_1).
I guess it would mesh with our instincts if discounting varied with age, but in the simpler special case where we consider only timespans that are short relative to our whole lives the theory works well; there's room to consider how this extends to a more general theorem.
No, that's too general; for instance, hyperbolic discounting is F(a,b) = b-a, but hyperbolic discounting is inconsistent in the relevant sense. For consistency we need F(a,b) F(b,c) = F(a,c), or equivalently F(b,c) = F(a,c) / F(a,b) = G(c)/G(b) where G(t) = F(a,t). (Note that the dependence on a has gone away.) This is equivalent to discounting things at time t by a factor G(t), which is the general form I described.
Depending only on time differences is the same thing as being invariant under time-shifting your whole life.
Akrasia is the tendency to act against your own long-term interests, and is a problem doubtless only too familiar to us all. In his book "Breakdown of Will", psychologist George C Ainslie sets out a theory of how akrasia arises and why we do the things we do to fight it. His extraordinary proposal takes insights given us by economics into how conflict is resolved and extends them to conflicts of different agencies within a single person, an approach he terms "picoeconomics". The foundation is a curious discovery from experiments on animals and people: the phenomenon of hyperbolic discounting.
We all instinctively assign a lower weight to a reward further in the future than one close at hand; this is "discounting the future". We don't just account for a slightly lower probability of recieving a more distant award, we value it at inherently less for being further away. It's been an active debate on overcomingbias.com whether such discounting can be rational at all. However, even if we allow that discounting can be rational, the way that we and other animals do it has a structure which is inherently irrational: the weighting we give to a future event is, roughly, inversely proportional to how far away it is. This is hyperbolic discounting, and it is an empirically very well confirmed result.
I say "inherently irrational" because it is inconsistent over time: the relative cost of a day's wait is considered differently whether that day's wait is near or far. Looking at a day a month from now, I'd sooner feel awake and alive in the morning than stay up all night reading comments on lesswrong.com. But when that evening comes, it's likely my preferences will reverse; the distance to the morning will be relatively greater, and so my happiness then will be discounted more strongly compared to my present enjoyment, and another groggy morning will await me. To my horror, my future self has different interests to my present self, as surely as if I knew the day a murder pill would be forced upon me.
If I knew that a murder pill really would be forced upon me on a certain date, after which I would want nothing more than to kill as many people as possible as gruesomly as possible, I could not sit idly by waiting for that day to come; I would want to do something now to prevent future carnage, because it is not what the me of today desires. I might attempt to frame myself for a crime, hoping that in prison my ability to go on a killing spree would be contained. And this is exactly the behavour we see in people fighting akrasia: consider the alcoholic who moves to a town in which alcohol is not sold, anticipating a change in desires and deliberately constraining their own future self. Ainslie describes this as "a relationship of limited warfare among successive selves".
And it is this warfare which Ainslie analyses with the tools of behavioural economics. His analysis accounts for the importance of making resolutions in defeating akrasia, and the reasons why a resolution is easier to keep when it represents a "bright clear line" that we cannot fool ourselves into thinking we haven't crossed when we have. It also discusses the dangers of willpower, and the ways in which our intertemporal bargaining can leave us acting against both our short-term and our long-term interests.
I can't really do more than scratch the surface on how this analysis works in this short article; you can read more about the analysis and the book on Ainslie's website, picoeconomics.org. I have the impression that defeating akrasia is the number one priority for many lesswrong.com readers, and this work is the first I've read that really sets out a mechanism that underlies the strange battles that go on between our shorter and longer term interests.