The most canon map is the one I've already posted. It gives the size in the far right, as well as the population.
Ah. A bit higher population than I expected, much, much smaller than I expected.
Your talking about a mere 50 light years to earth. And basically all the important bits of the federation are in that 50 light year radius sphere (Earth, Vulcan, the other founding members).
This contains only about
130 star systems with a bright enough star to be, well, nice to live around. Now, there are about 1,400 star systems, but a lot of those are brown dwarfs. They may have facilities, outposts, and other such things which make them not totally ignorable, but they're probably not likely to be major hubs.
The tiny number of places that need to be surveyed makes me think hyperspace really isn't going to be a huge problem: most of the jumps they're going to be doing are exceedingly close ranged: even before things are fully explored, just charting out a very conservative path (say, jumping to the edge of a star system, surveying where everything is, then jumping to the edge of the next one, excertera) is only going to be, well, maybe six days to earth?
Hell, the distance involved may be short enough that simple telescope and gravitational readings from the wormhole may be enough to figure out a direct rout to earth (its possible that there might just be a perfectly clear line of site jump that can be done to earth itself: with the distances involved there's only really likely to be maybe 1-2 stars in the way: there may simply be no large objects between earth and the starting point at all at such relatively short galactic distances).
With such a small area to explore, you could probably well map things out pretty quickly with one Star Destroyer fleet (a star destroyer plus support vessels) Especially if you equipped it with FTL capable fighters. Plus, well, Droid spam. With only about 200 star systems to really focus on, that's probably well within the capacity of a Star Destroyer to spam.
So, I think getting a clear hyperspace rout figured out isn't really going to be that big of an issue. The distances involved are just too short. Plus of course there's the issue that a lot of this sphere is already explored: its probably not going to be hard for the Empire to find someone with a fairly complete star chart: I mean, every random freighter who finds themselves in such an unfortunate position to be captured by the Imperials probably has a huge wealth of astrological information, which can serve as a major building block for Imperial mapping.
Star Fleet has basically zero strategic depth against Star War's hyperdrive here: the distances are near instantly transversable by the hyperdrive, and the distances are so short that nothing but local short range jumps are really necessary, and the number of targets that need to be surveyed is so small that whatever level of "field engineering" is necessary is almost certainly well within the capacity of basically a single Star Destroyer. The imperial bureaucracy approving the deployment of a team of, whatever they would be called, hyper space mappers? Is probably going to be more of a time delay for the Empire than actually figuring out a safe rout to Earth. Assuming this is beyond the capacity of the standard navigation team for a star destroyer, which I'm not sure would be a reasonable assumption.
Edit:
and ignoring the Federation interfering in various ways with the Empire performing a leisurely, throughow by the books star mapping. That could delay things further: destroying the drones before they get the chance to collect a lot of data, maybe hacking one to purposefully provide bad Intel, Destroying any explorer craft, messing with gravity and such, exectera.
Plus there does seems to be a difference between "charted rout" and "totally safe" routs. Maybe something like the difference between a well pathed and smoothed highway vs a quickly built dirt road through the forest.
Though normally things going wrong with the rout seem to result in the craft being pulled out of hyperspace, rather than destruction.
Edit 2: 8/16
Secondly, on the scale variation.
The Federation is listed above to have a population of a little under a Trillion. Coresant itself has a listed population of over a trillion. Coresant thus, as a single planet (admittedly the capital), has more people than the entire federation. A couple trillion is a reasonable number for a city planet, due to population density.
Coresant is not the only city planet in the Galactic Empire. A previous spacebattles thread said the EU Empire had a population estimate of about 100 quadrillion, or 100,000 Trillion. This is is probably a good idea of the scale difference between the two - 100,000 - 1. The economic scale is probably a little bit less lopsided in favor of the Empire, since there's a lot of poor areas of the galaxy, though an overall average is pretty difficult to actually determine.
But, the scale difference does make Federation resistance, difficult. The Empire wouldn't necessarily even have to do much: for example, on Earth, a fairly well controlled and survailed place, there were still some
200 pirate attacks. Lets say, for ease of numbers, that this involves a 1,000 people. Lets say this is a rate of 1,000 Pirates per 10 Billion people. If only this many people were pirates as a ratio in the Star Wars Galaxy, then this suggests you have 100,000 Pirates per trillion people, 100 million per quadrillion. So even based on our rate, there would be an estimated 10 Billion pirates in the Star War's galaxy. Given how much more lawless parts of the Star War's galaxy is, and how much easier piracy is, I wouldn't be shocked if there were nearly a trillion or more full time pirates, and an even larger number of part time ones (say, a salvage ship that is generally legitimate, but sometimes salvages something that's not quite free game).
So, as an example of what the Empire could do if they can't deploy enough forces to outright conquer, they could simply set up the beach head, and declare that everything outside their border outlaw: outside the law of the Empire. Let pirates and mercenaries strip the federation of valuable equipment, setting up an Imperial base to buy all the valuable things, and maybe set bounties for certain things: for example, if you learn the name of a brilliant scientist in some field your interested in, you can set a bounty and let a bobba fet or somesuch grad them for you. This way, Empire commitment is minimized, you draw some pirates and nardowells out to bug someone else, and get most of the main advantages of conquest anyways: the technology, extremely educated slave labor, more ships, exautic materials: the extent of the pirate activity is only limited by how profitable piracy can be made in the Federation.
You can even use this to go the Norman option if the piracy and ambitious warlords are damaging and demoralizing enough: keep open the possibility to bring planets from "outlaw" status to "inlaw" status: at the lowest level, extracting protection money from planets, at best getting them to agree to annexation. Try to negotiate planet by planet instead of federation wide.
Occasional attacks into Federation space, say limited one system offenses, would further degrade the federations ability to prevent the pirates from threatening federation worlds, drawing in fleets and assets off anti piracy patrols to fight the major battle.
But, once again, extracting tributes and annexing a couple of worlds will net probably 90% of the benefit of outright conquest.
And you can add to this another thing that the Star War's galaxy can do thanks to the immense disparity: just flood it with migrants. Many of the Federation worlds have such low populations that outnumbering them with new colonists would be pretty trivial to achieve. This would help to undermine Federation unity and complicate its ability to control its own space.
All of these are the long and slow options that seem pretty impossible to fight effectively. Of course, that "long term" is going to be a bit of a problem: unlike the Muslims or Romans in spain, who used similar tactics to conquer that difficult terrain over a hundred or so year campaign, the Empire doesn't actually have a 100 years to slowly annex the area through a long campaign of piracy and steady subjugation.
