Is there a name for this pentatonic scale?
Well, the black keys themselves don't necessarily make a unique scale by themselves.
To show you what I mean, all of the black keys are in the B major scale. However, the same is also true of the F# Major scale (or the Gb Major scale if you like to use flats instead). So it kind of depends on the context you're using them in.
However, since you mention a pentatonic scale, I'm assuming you're interested if there is a scale that has just the 5 black keys in it, and nothing else. In that case, you're in luck. The Gb Major Wikipedia article has something interesting on that:
In particular, the black keys G♭, A♭, B♭, D♭, and E♭ correspond to the 5 notes of the G-flat Pentatonic Scale
So the name you're looking for is probably the Gb Pentatonic Scale.
[W]hat's the best term to refer to the black keys?
Again, it really depends on the context you're using it in. However, I think "scale" is probably the appropriate term here. Actually, "pentatonic scale" is probably the more appropriate term here since there are only 5 notes in the scale.
I'm not going to re-hash here the differences between "modes", "scales", and "keys". The other question you reference does a good job of explaining the differences in my opinion (or at least it does a better job at explaining them than I could do explaining them here). If you are still confused after reading that question, perhaps it might be a better idea to ask about that confusion in more detail in a separate question.
I notice that when I'm doodling on them, no matter what note I start with it often likes to come to rest on G♭; that seems to be a tonic note. Why?
This to me says that, again, you're probably using the Gb Pentatonic Scale. Gb is in fact the tonic note of that scale, and that's probably what your ears are picking up on.
What are the rules/guidelines for harmonising with black key melodies? When I listen to other music using the black keys, the chords are seldom purely diatonic but are there useful guidelines for writing chords to harmonise black key melodies?
That's probably too broad of a question to do justice here, but the short answer is probably the same as with white keys.
Music theory in general doesn't usually differentiate between "white" and "black" keys; music theory usually deals with scales. That is to say that it doesn't really matter if you are using white keys or black keys, but rather if you're using a major scale, minor scale, pentatonic scale, or other type of scale.
So, what are those rules exactly for each of those types of scales? I couldn't tell you. I barely have a handle on major scales; I certainly couldn't tell you about rules for minor or pentatonic scales. That's just knowledge I don't currently have. But I can tell you that type of scale probably makes a lot more of a difference than if the keys are white or black.