Instead of using the proper counter for each thing, is it okay to just use the つ counter?
Like in:
彼女は子猫2匹を持っている。
Use つ instead of 匹:
彼女は子猫2つを持っている。
Are there situations when this is incorrect or not recommended?
Instead of using the proper counter for each thing, is it okay to just use the つ counter?
Like in:
彼女は子猫2匹を持っている。
Use つ instead of 匹:
彼女は子猫2つを持っている。
Are there situations when this is incorrect or not recommended?
つ is a generic inanimate counter. It might be confusing sometimes, but you can get away with using it for lots of inanimate objects if you don't know a more appropriate counter.
Don't use つ for animals or people.
You should use 匹 for kittens in your example, not つ. The basic counter for people is 人{にん}.
In general, animate objects (animals and people) have special counters. No matter how long and thin a snake may be, you can't use the inanimate counter for long and slender objects (本) to refer to it. And you should never refer to people like they're inanimate objects.
If you don't already know them, you should try to learn some basic counters at some point. Planning on using つ all the time is no good. Some of the most common ones include 人・名・匹 (animates) and 本・枚・台 (inanimates), along with つ and 個.
It might work for most objects.
For some other concepts it won't be understandable at all, though, for instance don't use it for: