According to Wikipedia, the Orion Arm of the galaxy is 10000 ly long, by about 3000 ly wide and 1000 ly thick. Therefore, it would seem reasonable that this volume of space can contain the entire Federation and all its neighbours. - Assuming high warp speeds based on Voyager allowing travel of approx 1000 ly per year (~3 ly/day) the arm would take 10 years to travel the length of. So, given the relatively low travel times within trek stories, one would have to assume that, with the exception of stories involving super fast / long distance travel, everything does happen in the Orion Arm. Limted exploration or survey work may have been undertaken in the Saggitairus or Perseus arms, but these seem to be at least 5000 ly distant. This still leaves the 'above or below' argument with regard to the dispersion of the major empires in ST. THis would also allow the Enterprise in WNMHGB to pass 'up' or 'down' through the galactic disc and reach the edge of the galaxy in approx 6 month's travel, at VOY speed.
FWIW, it would seem that Voyager started in the Carina-Saggitarius arm, then crossed to the Scutum-Centaurus arm (via the Nekrit Expanse?). Borg space is generally located in the Scutum-Centaurus arm. Later parts of the journey, possibly undertaken with the quantum slipstream jump, would take Voyager into the Norma Arm, where S5-7 take place.
The Dominion would seem to be located in the farthest part of the Scutum-Centaurus arm also.
Alpha quadrant would be top right
Many of the Federation's neighbours seem to be based on military expansion and would likely envelop systems inhabited by other, less advanced races, only stopping if communication / supply lines become too long, or they border another empire of comparable strength. It would herefore be reasonable to assume that these empires generally occupy contiguous bubbles of space. the Federation, OTOH, expands through treaty and colonisation of uninhabited systems, which as I'm sure I have seen discussed before, will lead to a far more irregular shape, including pockets around independant worlds. I would envisage the Federation being shaped a bit like a cartoon splat shape, but in 3D, perhaps with some exclaves, and a few enclaved areas of independant 'nations'. This would allow the Federation to be big, yet also border a very large number of races, and also for the distance between, for example the Klingons and Cardassians to be relatively low. Additionally, until the treaty was broken, the alliance with he Klingons would presumably allow them free passage, no questions asked. Consider maritime law and law of the high seas, and the rights of travel associated with.
Blueshirt, with regard to your reference to the 'arbitrary' position of the Greenwich Meridian, only a prime meridian in western Europe allows for the date line to be placed entirely within ocean (albeit with a few meanders here and there). A prime meridian, for example off the westernmost part of Africa (as I believe was once considered) would leave the date line crossing eastern Russia and Australia.