Every mainstream Pokémon game (i.e. every handheld iteration) since Ruby/Sapphire uses the same RNG, with slight variations on how they seed, cycle through spreads, and so on. So yes, every fourth generation spread should be technically available within FireRed/LeafGreen (though I am not sure how method K works exactly, I believe it is just a restriction of method 1). However, we do not know just how exploitable the RNG is at this point; how actively we can control the seed. For instance, in Emerald, we cannot change the starting seed at all (which, incidentally, is what makes it so easy to manipulate). As a result, it is "technically" possible to obtain a 31/31/31/31/31/31 spread, but one would have to wait weeks (even months depending on the spread) for it to roll around, and hope that the spread would even be hit on that attempt!
We know that FireRed and LeafGreen do change their seed, and that the same seed can be reproduced through consistent timing (see chrish's research). However, we do not know exactly how to alter this seed to a specific result, as one enters a certain time in Diamond and Pearl. It may be that we are limited to a certain number of seeds, thus leaving only certain spreads within practical reach. If Emerald is left on for a year or so, all spreads will be cycled through; presumably FireRed/LeafGreen have more spreads easily obtainable because of a diverse seeding process. However, to answer your question, I believe that it would suffice to say that any spread in a fourth generation game is theoretically obtainable in FireRed/LeafGreen. More difficult, perhaps, to obtain, but theoretically obtainable.
As for your question on the state of this research. I cannot speak for any individual member in this thread, save myself, but I believe that the research is slow, but ongoing. We still do not know how the game seeds (any help is appreciated!), but if you look at the different posts in the thread, I think you will find that we have made progress from October, when it was thought that obtaining the same seed was impossible. However, on a personal level, I am far less involved in this research than I would like to be; it is due to my winter holidays that I am even posting right now. However, this is due to schoolwork eating up my time; as I posted earlier, I encourage everyone to continue posting their results.
I thank you, among others, for showing interest in this undertaking. An even bigger thanks to all those who have contributed research up to this point! This will be my last post on this subject for 2009, but I hope that this endeavour will stay alive into the new year.