I know this sometimes が can be replaced with の, and as far as I know this happens when a noun is modifying a noun.
Is this all there is to it?
Is it important to always replace が or is it optional?
Can it always be done?

As far as I know, "no" can but doesn't have to replace "ga" in subordinate clauses. Sorry, I don't have a useful keyboard at hand for Japanese text at the moment.

As bertoni says, I think the situation you're referring to is when an entire phrase modifies a noun, and the subject of that phrase is marked with の instead of が.

足が速い人
足の速い人

人がよく来る映画館
人のよく来る映画館

I think it's mostly a matter of preference and either is acceptable.

In classical Japanese, が and の both had pretty much the same function (が was also used for possessives) and that may have something to do with it. One other theory I've always held, though I don't know if there's any truth to it, is that the subordinate clauses could be broken down two ways and the の conveys that everything after the の is something that "belongs" to the thing before the の:

[足の速い]人
足の[速い人]

[人のよく来る]映画館
人の[よく来る映画館]

I've never truly understood clauses to be honest, so just out of curiosity. Is this kind of an example where the two can be switched?

私によって金がある人はむかつく
私によって金のある人はむかつく

This is probably ungrammatical but I just thought it up on the spot. Is the 金がある・金のある the kind of "subordinate clause?"

Yes, I'd say that's correct.

A subordinate clause is just an embedded sentence, in this case, so I agree that the "ga" and "no" are both fine in your example.

From DOBJG's explanation of relative clauses, p. 378:

3. No can also be used to mark the subject in relative clauses, as in (6).

(6) ジョン が／の 食べたステーキ
(the steak that John ate)

No cannot be used in place of ga, however, if the subject of the sentence is followed by a noun, as in (7a). If no is used in this construction, the meaning of the sentence changes, as seen in (7b).

(7) a. トム が フットボールの切符 を あげた女の子
(the girl to whom Tom gave a football ticket)

b. トム の フットボールの切符 を あげた女の子
(the girl to whom (someone) gave Tom's football ticket)

I read before that の puts the emphasis on what follows it, whereas が puts the focus on the subject like it usually does. I suppose in this way it has similarities with the difference between は／が, but it doesn't seem to be anything like as strict.

Sometimes using のsounds quite natural to me because it almost sounds like possessive の, e.g. 彼の言うこと. 彼が言うこと still has 18-million hits on Google though, so I don't think this is anything to worry about too much, as long as you're able to understand it.