sa chatte possible?
Yes it is possible.
Male cat - son chat
Female cat - sa chatte
These don't change if it is a man or woman calling.
april 13 2018 marked wrong for 'sa chatte'
August 20 marked incorrect for "sa chatte"
It is grammatically possible. However, if you don't know the gender of the cat, you use the default masculine form (it's true). We don't know the gender of the cat and I presume that you don't either.
I'm guessing he's making the call because the cat cannot make the call?!
Not "en train" ?
The expression "être en train de" (+ infinitive) is only used to emphasize the fact that the action is occurring at this very moment. So you could say, "il est en train d'appeler les pompiers pour son chat". And yes, it is accepted.
Why not Il téléphone...
"Pompiers" ? Those who pump (water on to fires)?