What is your physics specialization? Foundational cosmology? I didn't pursue my PhD after my masters because I couldn't decide on a specialization. I thought particle physics would be for me, but general relativity was infinitely more interesting. I've always liked seeing the connections between ideas and the overall picture...even biophysics has its topics of interest. Anyway, perhaps you've had more cosmology/general relativity than me and can clarify, what do physicists mean when we talk about the age and size of the universe? Is that a quantity that all observers will agree upon regardless of their reference frame? If so, wouldn't that age and size serve as an absolute clock or an absolute measuring rod? To me its no suprise that the assumptions leading to the Robertson-Walker metric would lead to an origin or a "special point" of the universe in time...we assume spacial homology and spacial isotropy but assume a unique "forward" direction in time. Its like assuming a unique "up" direction on the surface of the earth and then being suprised that we have a spacial center in 3-D space. Personally, I think the big bang theory has a major overhaul in store during this century anyway...so I'm not too worried about the beginning of the universe being in conflict with its ontological necessity.
You know, I've never really thought about that. No, observers wouldn't agree on the size of the universe in different reference frames -- though by "size of the universe" we can only talk about the visible universe. An accelerated observer will actually end up with a cosmological horizon behind them as light fails to be able to catch up to them, and in fact they'd experience an increase in heat from "hot" photons. I can't remember the name of the effect, I think it's Unruh or something. (Edit: yes, it's Unruh: http://en.wikipedia.org/wiki/Unruh_effect)
Essentially the cosmological horizon behind them means from their reference frame they lose some of the information about photons in their vicinity, making their motion appear to be random and therefore their thermometers would go up whereas a relatively stationary observer wouldn't observe the same -- and be able to see more of the universe than the accelerated observer.
If we try to ask whether the entirety of the universe is the same size in all reference frames I'd really have to think about that. I'm already aware of one problem with these types of questions: for instance, the Planck distance. Either the Planck distance is absolute or it's not; QM says it is, relativity says it depends on your reference frame. Seems to me like the reverse of the paradox from the ultimate small to the ultimate large.
As for the "forward arrow" in time it's a consequence of deep thermodynamics... which are themselves a consequence of sheer probabilities. The RW metric is mostly just based on homology and isotropy; the only factor/assumption that bothers with an arrow of time is the scale factor a(t) which is based on relative expansion as far as I know. We preeeeetty much just sort of plug in what we gather to be the "start" of the universe based on our other methods for calculating the age of the universe, so really the metric doesn't make a statement on there being a special "time" -- we arbitrarily choose those times and plug them in. In this case we just sorta say that the BB event is the "start" even if it doesn't mean it's the start of de facto existence altogether.