It's hard to explain but you can leave out с (with) where the thing is an instrument by which you do something... like when you write with a pen, eat with a fork, etc, you don't use the article. This soup is not made "with" potatoes and onions in that sense... it's made with a pot (no article), and the potatoes and onions are things that are added to it (you always need the article in such instances).
It is true. Russian has variant instrumental forms for feminine nouns, она, and adjectives modifying feminine nouns:
- с женой / с женою
- такой/ такою, хорошей/хорошею
To my ear, the two-syllable option sounds fairly dated and bookish for most words, though some speakers do use ею as an instrumental of она. In general the -ю endings are rarely used. For instance, картошкою appears 10 times in the Russian Corpus, seven of of them 50 years ago or older. Картошкой has over 1000 instances.
You may want to use one of these in poetry, though. Sometimes you need that extra syllable.