1. Is the top node of the tree (terminal?) able to c-command anything? I have been using the 'up and over' method for c-commanding, but obviously, with the top node there is no way to go 'up'. Does this mean it does not command anything, or does it command everything?
Yes, up-and-over is a good way to think about it.
Your question about the top (=root*) node applies to any phrase/constituent and its components. It's not clear to me whether in principle c-command should apply in those cases (probably not by definition), but it's effectively irrelevant: having something c-command part of itself doesn't seem like a very useful idea, so it just doesn't really come up, aside from completely abstract questions regarding labeling like "does node A c-command node B?". Think of c-command as applied to binding theory: how would a whole phrase bind part of itself? We can try to work around some awkward phrasing that might be like that as in "The students sitting by themselves", but that gets into very complicated other issues (regarding silent pronouns, non-finite clauses, etc.) and generally involves crossing a binding domain anyway, so something else would be going on. Remember that when we're talking about one node c-commanding another, we're really talking about the whole phrase/constituent contained within that node, not just the top node itself. So your question would be the same as asking "can a whole sentence c-command part of the sentence?" which just doesn't seem to really apply in a relevant way. (But it's great to
ask such questions to explore how c-command works!)
[*root = first/top node in a tree, that dominants all others; a "terminal" node is one of the bottom nodes, i.e. a word/morpheme, rather than intermediate level in the structure]
A more interesting question to explore might be looking at "backwards" c-command/binding, which is just normal c-command but could apply to an anaphor that occurs before its "antecedent". That's somewhat limited in English (due to word order), but you can try to find some examples yourself. There's nothing restricting the direction of c-command (which is why it is preferable to more intuitive ideas of precedence, which don't work out as well when we look at the full range of data).
2. What is meant by the 'category' of a binding domain? Is this TP, VP, DP, etc. If so, the way I have been taught the binding domain for Principle A is that it is the closest TP that contains the VP that is being bound, does this not make all binding domain categories a TP?
Defining "binding domain" is actually very difficult. A rough approximation has to do with clauses, and that works most of the time for typical uses. But there are some other cases that must be treated a little differently, and it doesn't seem to be strictly TPs. The textbook I use approximates this in a different way: it's any XP (i.e. any phrase) that
has a subject, which is most often a TP with a subject, but in principle might be any other phrase with a subject or specifier position. If you're familiar with so-called
small clauses (
https://en.wikipedia.org/wiki/Small_clause), you may already know that other phrases can have "subjects", as in for example a VP small clause with a subject:
I heard you sing, where "you" is the subject of the VP sing (in a position just like a TP subject, but within that VP).
Another case where this works out nicely for demonstration purposes is with possessive DPs, assuming that the possessor is actually a "subject"/specifier within the DP, and the head of the DP (=D) is the
's marker of possession. This analysis isn't immediately intuitive, but works well (although it might not be included in your textbook). Here's an example:
[tree][DP [DP [D the][NP [N student]]] [D' [D 's] [NP [N book]]]][/tree]
Now, think about binding: note that the DP "the student" c-commands the NP "book". So far, there's no boundary for a binding domain, because there's no subject position between them. But if we expand that a bit, we get an interesting sentence:
John saw Mary's picture of herself.*John saw Mary's picture of himself.The reason the first sentence is grammatical is because
Mary c-commands the whole NP "picture of herself",
including the embedded DP "herself". (Try to draw a tree to see how that works, but it's similar to the example above.)
The second sentence is ungrammatical because even though
John c-commands
himself, they are not in the same binding domain: the possessive DP with
Mary as subject intervenes, meaning that a new binding domain is established.
So one way to think about binding domains is that they are formed by any XP that
has a subject. This is still a rough approximation in some ways, and we could explore even more data to refine the idea. But it's clear that it's not only TPs that can potentially form binding domains. There's more you could read in published articles, maybe more advanced textbooks, etc., but that is probably beyond what you're expected to know for an introductory class. (Again, asking questions like this is great, because you learn more as you do so, and you better understand the basics by looking at the advanced cases. Actually that's how science progresses in general: establish a theoretical explanation for the basic cases, then refine it by looking at more and more data.)
There are also some other more advanced topics about binding and binding domains, such as so-called "long-distance anaphors" in some languages (where a word meaning something like "himself" or "herself" can be used across a clause boundary), as found in some other languages (not English). That's been an interesting area of research challenging and/or expanding Binding Theory in general.
