The ever confusing "that and "which"

The fast and dirty rule is that if it's in a clause that the sentence doesn't need to retain meaning, use which. I think of it as a matter of specifics that for specific things, which for things unspecified. In your sentence, we are talking about a specific coat, the one that very nearly covered his boots. I think you want 'that' there.

In your first bike sentence, the word 'that' isn't needed at all.
In the second bike sentence. I think it should be 'that' because it's the specific bike that his mother bought. Which one? That one his mother bought.

Rabid wombats, which sometimes bite people, are not to be messed with. --The clause is not needed for the sentence to have meaning, and we aren't talking about a specific wombat.

A rabid wombat, that mean one Carl poked with a stick, is now chasing us. --We are talking about a specific wombat.

Now, all that said, wait to hear from a true grammar superstar, because I may have this completely backwards.

Benedetto Youssef said:

I understand that "that" is to be used for a restrictive clause. Example: He rode his bike that his mother bought for him to the store. (multiple bikes?)

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Technically that's correct, but a cleaner, smoother version of that sentence would be: He rode the bike his mother gave him to the store.

But what of a sentence like this: He wore a dark brown fedora hat and a long black trench coat that nearly covered his brown leather boots.

Is there anything wrong with my usage of that (following long black trench coat) in this sentence?

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Not at all. It's perfectly correct. And it's also a good example of when "that" is necessary in order for the sentence to make sense (unlike the sample sentence above, where "that" wasn't needed).

Bear in mind that there are differences in usage between American and British English, which can further contribute to the confusion.

ETA: just realised the above sentence has "which" in it, which Americans may prefer to be "that".

ETA#2 - lots of "that"s can be safely removed from sentences without changing the meaning, and they usually flow better without it.

You're pretty much there with the restrictive and non-restrictive clauses. What muddies the waters a little is that, in practice, it's not always such a clear line between the two.

Here's how I understand it:

Think of restrictive ("that") clauses as making the meaning of your sentence more specific -- they "restrict" the meaning. That doesn't just mean restricting it to a particular object (as in your bike example). I can also be used to add essential information about an object in order to give your sentence a narrower meaning.

Non-restrictive ("which") clauses, on the other hand, provide non-essential additional information about the thing you're discussing. The information could be identical to what you might otherwise include in a restrictive clause -- the difference is that in this case, it's not essential to what you're trying to say.

For example, I could say:

"He was wearing a fedora, which looked like it had been stolen."

Or I could say:

"He was wearing a fedora that looked like it had been stolen."

In the first, I'm telling you that he was wearing a fedora. Additionally, that fedora looked like it had been stolen. The fact that it looks stolen isn't a core part of the sentence, I'm just providing it as additional information. If you removed that clause, the core meaning of my statement -- that he was wearing a fedora -- would remain.

In the second, I am telling you specifically that he was wearing a fedora that looked like it had been stolen. I'm not just trying to tell you what type of hat he was wearing -- I'm trying to tell you something more specific, and if you removed that clause my meaning would not be the same.

In your case, the length of the trench coat is not being presented as incidental information -- it's a core part of the statement that you're making about him, and so "that" is correct.

Does that make it any clearer?