How should I properly use the word vice versa in writing? Is that even the correct spelling?
"Vice versa" is surely correct, see e.g. Merriam-Webster. Wiktionary suggests that you could write "vice versâ" if you wanted to expressly mark the ablative case, and they even provide a quote, but that is actually the first time I see that spelling in an English context (we don't have the ablative case, after all).
If you don't like the expression itself for some reason, you can go with the English equivalent "the other way round".