Hi, Elaine! Sorry I took so long to respond to this, and I'm not sure whether you're interested in learning Malayalam right now, but I hope this helps anyway.
Elaine wrote:I'm a man. - ഞാൻ ഒരു മനുഷ്യൻ ആണ്.
This sentence is fine, but you can also omit the copula ആണ്. So you can say either ഞാൻ ഒരു മനുഷ്യൻ ആണ് or just ഞാൻ ഒരു മനുഷ്യൻ.
I have two cats. - എനിക്ക് രണ്ടു പൂച്ചകൽൾ ഉണ്ട്.
You can also just say എനിക്ക് രണ്ടു പൂച്ച ഉണ്ട്.
She is a woman. - അവൾ ഒരു സ്ത്രീ ആണ്.
I'm not sure whether native speakers would accept this sentence. സ്ത്രീ 'woman' is a pretty formal word, whereas അവൾ is an informal pronoun that often implies that the speaker feels a certain amount of contempt for the person they're talking about. Probably a gender-neutral pronoun like അവർ would be more appropriate in this context.
I speak Malayalam. - ഞാന്ൻ മലയാളം സംസാരിക്കുന്നു.
I think this is a possible interpretation of that sentence, but it also means "I am speaking Malayalam (right now)" or "I am conversing in Malayalam." Usually, I think Malayalees would say "I know Malayalam," i.e. എനിക്ക് മലയാളം അറിയാം.