My stab at it:
a) I don't know how good Mounce's basics is. As you can well understand yourself, since we're talking about "basics", you'll probably need at least one other textbook to get into more advanced stuff, even if Mounce's is the best around. But I'm sure I'm not telling you anything you don't know already so let's move to
b) Will knowing Koine help you with Greek of other periods? Depends on what you mean.
i. Homeric Greek (that is, if you are interested
If not, it's not a crime/cardinal sin/whatever. De gustibus .... Not so much. Yes, it will help, in a structural level, but be prepared to be bewildered by the vocabulary at first. Apart from that, Homer tends to spell things out more, than, let's say John so, from a very specific and rather narrow point of view, it'll actually be easier than his gospel or some of the Epistles
ii. Attic/Ionic/Doric/Aeolic Greek (of which the most prominent is, of course, Attic). Will it help. Yes, most definitely so. Not without some extra reading for specific grammatical phenomena and types that where simplified by the time of Koine.
I also don't know how interested you are in using the "correct" reconstructed pronunciation. If you are and you are not using the standard reconstructed one which is used (incorrectly) for any form of Greek that can be called ancient as opposed to the reconstructed Koine one, then, well, you'll have to learn the new one. Hardly anyone bothers though but I thought I'd mention it.
iii. Hellenistic. Almost the same as ii but not so much. Way closer to Koine, especially if we're near or at Roman times.
iv. Byzantine Greek. There are some writers who preferred to write in Attic Greek. Apart from those, with knowing Koine, you are golden. You may have to check some vocabulary, especially when it comes to the titles of officials and such but that's it. If you have a smattering of Latin that becomes a child's play.
v. Modern Greek (I know you didn't ask but I add it for completion's sake
) Yes it will. It's more simplified than Koine, but you'll have to learn a thing or two. Also, depending on the pronunciation scheme you went for, you may need to study some on that field too (very close to reconstructed Koine, a rather long way from Attic and such).
In a nutshell; Any introductory book means you'll have to study extra. Knowing Koine helps a lot with other forms of ancient Greek but it's not as if you can just pick Plato and read through.
Anyway, that's how I see it.