Yes, the 'default' words for "kot" and "pies" are grammatically masculine.
I have a female cat, called Zuzia. Sometimes I talk about her in masculine (Kot jest głodny = The cat is hungry), when I just use the word "kot", and sometimes in feminine when I use her name (Zuzia jest głodna = Zuzia is hungry).
To make it clear that a cat is female, the word for a female cat is "kotka".
And the word for a female dog is "suka" or "suczka", but those have exactly the same negative connotations as the English word for a female dog.
Normally, Polish grammar doesn't make such distinction. It doesn't matter if you do something regularly or at the moment, you use the same verb. There are two exceptions to that:
Some verbs do have habitual forms that you can use if you want, but they are not really required. Here, you could say „Ten pies jada mięso” to mean "This dog eats meat" as opposed to "This dog is eating meat". „Ten pies je mięso” could mean either "This dog eats meat" or "This dog is eating meat".
Verbs of motion have both habitual and continuous forms that are required. You will learn the difference in a later lesson.