If you said this had been written by a native speaker, I would have believed you.*
Your thinking that you need all those she’s and would’s in order to be clear is exactly right. Clearly you are using grammar to communicate, by empathizing with the listener’s need for grammatical cues in order to understand the sentence. That’s excellent. You’ve even got nicely contrasting tenses in money she would get and cash she had been saving. It’s done so simply—that is, it exploits English grammar so effectively—a reader hardly notices the complexity of what’s being described.
The rhythm of the sentence is unusually good. Most native speakers don’t write sentences that sound this nice! The line sounds almost like poetry. It strikes a nice balance between repeating and varying its rhythms, and the recurring rhythms are in sync with the grammar. Notice that money and cash both land on accented syllables. The word herself is unnecessary, but it aids the rhythm—excellent choice.
Some of the other answers suggest ways to shorten the sentence, but I think the repeated use of would contributes a lot to that nice rhythm. It emphasizes the flow of events in Zoë’s life. The shorter wordings emphasize the house and the money, and they break up the rhythm.
Here’s how I’d tune it up just a tiny bit more. This version adds a that after decided so you get the same rhythm to occur twice at the beginning of the sentence, which fits the rest of the sentence a little better. And it replaces been saving with saved, to make saved better echo the verbs in the other “she”-clauses, and to make the contrast between tenses even stronger. I've marked the phrases with same grammar and rhythm in bold:
Zoë decided that she would sell her downtown apartment, and with the money she would get and the cash she had saved, she would buy herself a house in the suburbs.
If you’re looking for a generally applicable principle about repetition of grammatical words, it’s: repetition is good as long as each repetition varies something from what came before and each repetition brings progress. This is an aesthetic matter, so that’s not a strict rule, of course. But one reason the sentence works so well is that each repetition describes a different time, and the repetitions lead straight to a conclusion which comes latest in time as well as last in the sentence.
Seriously, if most of the sentences you write have this excellent parallel flow of rhythm and grammar, you should be writing English professionally.
* Except that apartament should be apartment.