Asimov's description is pretty much correct. There aren't many stars out there, so the night sky away from the galactic disk would be fairly dark. Toward the galaxy you you have an edge on view of the galactic disk.
As for a sky full of galaxies, you might see a few but probably not. They are intrinsically very faint and moving a few tens of thousands of light years wouldn't do much to make them closer and therefore brighter. The sky background would be lower so you might be able to see a few more but most likely, to the unaided eye it wouldn't be much different.
So if you looked away from the galactic center, there would be a very darks sky with only a few stars and a few faint smudges just on the edge of detectability for your peripherial vision.
The estimates I've read are similar to yours: 200 to 400 billion stars. Counting the stars in the galaxy is inherently difficult because, well, we can't see all of them.
We don't really count the stars, though. That would take ages: instead we measure the orbit of the stars we can see. By doing this, we find the angular velocity of the stars and can determine the mass of the Milky Way.
But the mass isn't all stars. It's also dust, gas, planets, Volvos, and most overwhelmingly: dark matter. By observing the angular momentum and density of stars in other galaxies, we can estimate just how much of our own galaxy's mass is dark matter. That number is close to 90%. So we subtract that away from the mass, and the rest is stars (other objects are more-or-less insignificant at this level).
The mass alone doesn't give us a count though. We have to know about how much each star weighs, and that varies a lot. So we have to class different types of stars, and figure out how many of each are around us. We can extrapolate that number and turn the mass into the number of stars.
Obviously, there's a lot of error in this method: it's hard to measure the orbit of stars around the galactic center because they move really, really slowly. So we don't know exactly how much the Milky Way weighs, and figuring out how much of that is dark matter is even worse. We can't even see dark matter, and we don't really understand it either. Extrapolating the concentrations of different classes of stars is inexact, and at best we can look at other galaxies to confirm that the far side of the Milky Way is probably the same as this one. Multiply all those inaccuracies together and you get a range on the order of 200 billion.
Best Answer
Yes, everything that appears as a point like star is in the Milky Way. The most nearby stars outside of the Milky Way are in the dwarf galaxies that are Milky Way satellites, such as the Large and Small Magellanic Clouds. These appear as fuzzy little blobs to the naked eye, just as Andromeda does.
The only exception to this that I can think of is when a supernova occurs in a nearby galaxy. The most recent supernova visible to the naked eye was 1987A, which occurred in the Large Magellanic Cloud. Supernovae in Andromeda could also be visible to the naked eye as point sources.