How important is gender in Welsh?
Rather important, such as for mutations.
Darn, I just typed down a bunch of nouns in Welsh but didn't include gender :/.
A quick tip - just mark feminine nouns with a (b) (for benywaidd - feminine) and don't bother to mark masculine nouns. Roughly two-thirds of Welsh nouns are masculine, so you can save a little writing by only marking the feminine ones.
A few nouns are typically plural when you usually use them (tatws - potatoes; bresych - cabbage/s) - mark those with the letter (ll) (for lluosog - plural).
I actually noticed most were masculine, but I won't write the gender for the masculine nouns anymore. Words like tatws and bresych have singular forms don't they? Like isn't it taten and breyschen?
When learning nouns in any gendered language, you should always learn the gender with the noun as a single unit.
It is simply part of the language, that's all, just as in French, German and many others.
People makes mistakes in gender, even when fluent. If you do get something wrong in speech, though, just carry on, as your meaning will be generally be clear from the context.