Whether a relative clause is defining or non-defining (some writers say integrated or supplementary) is really more about semantics than punctuation. The first relative clause in the Original Poster's example is probably just an unconventionally punctuated non-defining relative clause. The clause neither explains which money it was that was given nor is an integral feature of the money that was given. Rather, it is a supplementary story about what happened to the money after it was given. (However, see further below)
The Original Poster's feelings about sentence (3) are actually the feelings that we should have about sentence (1). If this was actually a defining relative clause it might give us the uneasy impression that the money was already spent at the time of giving.
The rules for the omission of a relative pronoun can be summarised as follows:
- The clause will usually be a defining relative clause.
- The pronoun may not be the subject of the main clause in the relative clause (in technical terms it may not be the subject of the matrix clause in the RC).
- The relative pronoun may not be the complement of a preposition in a fronted prepositional phrase.
The second rule there means that it is not important whether the word is the object of the clause or not. The only thing that is important is that it isn't the subject of the main verb in the clause. To illustrate the point, the pronoun may well be the subject of a subordinate clause within the relative clause:
- That is the man [who] we suspect took the money.
In the relative clause above, who represents the subject of the subordinate clause:
The Original Poster's Questions
We can, of course, say (1). However, we would need a special reason to present the relative clause as a defining relative clause. We need some reason for which she spent immediately to be an integral part of the money. Here's a contrived example:
A: People never spend the money you give them immediately.
B: Rubbish! I gave Jane some money she spent immediately.
We could of course say money which she spent immediately and include the relative pronoun. This sentence does not quite have the same meaning as the Original Poster's number (3). The reason for this is that this sentence presents the fact that the money was spent immediately as an integral feature of the money for the purposes of the conversation.