I can't quite make it out from the TTS: is the "dd" pronounced? Or is it just dropped, since it's very similar to the preceding "th" sound?
It shouldn't be, but the audio definitely sounds like cath u, rather than cath ddu....the th and dd are not the same sound, so you ought to sound/hear both of them
Are mutations the only way to find out if a noun is masculine or feminine?
They're a good way. Another would be that feminine singular nouns take a soft mutation after y "the", so y gath "the cat", y ferch "the girl", y broblem "the problem". One thing to watch out for is that ll and rh never mutate after the word y: y llygoden "the mouse", y rhaw "the spade" - both of these are feminine.
The above rule applies to un "one" with feminine singulars too: un gath, un broblem, un llygoden, un rhaw.
If you see the words dwy, tair, pedair, these are the feminine forms of dau, tri, pedwar - "two", "three" and "four". So you may see dwy gath, tair problem, pedair llygoden.
Other ways of determining gender are categories (liquids are masculine, clothes are usually feminine) and suffixes (masculine -yn like pysgodyn "fish", feminine -en like llygoden "mouse".
You can read all about how to determine noun gender here: https://m.reddit.com/r/learnwelsh/comments/4k83ww/the_ultimate_welsh_noungender_reddit_post/