Yes "Pedwar" is wrong here because they're not interchangeable. "Pedwar" goes with masculine nouns whereas "Pedair" goes with feminine nouns.
So in the context of money will 4 always be feminine? Are all numbers feminine with money?
No, the fact that it's money has nothing to do with it needing the feminine form of "four". The nouns "Punt", "Ceiniog" and "Doler" (Pound, Penny and Dollar) all happend to be feminine, and therefore all need "Dwy", "Tair" and "Pedair", but the noun "Ewro" (Euro) is masculine and therefore takes "Dau", "Tri" and "Pedwar". Also both the words for "Money" (Arian and Pres) are both masculine although you'll probably never be using them with numbers.
That's a difficult question to answer. Borrowed words tend to be masculine, but not always, and that tends to be the up to the sway of general usage; if most speakers are treating a word as masculine or feminine there's little point any dictionary or regulatory body trying to claim to the contrary. Of course some words can be either, both "Tafarn" and "Munud" (both being borrowed along time ago from latin- the second potentially through English) can be treated as masculine or feminine varying due to dialect or even personal preference. Otherwise if the word is newly coined from pre-existing Welsh parts then it will probably follow any trends in ending e.g. "cyfrifiadur" ends with "-iadur", where "-iadur" words tend to be masculine. Most new words tend to fit into these categories of borrowing or formed using native suffixes which can hint to what the most natural sounding gender will be. For example if we take "Hunlun" (selfie) and break it into "Hun" (Self-an ungendered reflexive pronoun) and "Llun" (Picture-a masculine noun) it makes sense that it'll be masculine. (Although the ends-with rule that exists in German i.e. that since "Buch" is neuter any words ending with "Buch" are automatically neuter as well, is not necesarily the case with Welsh.)