Hello! I've been studying Japanese for a few years and I have a question about the use of "だ"

I originally thought it was merely an informal です, but was told that I shouldn't think that way or it'll cause many errors in my Japanese. I also previously thought that it was used in written Japanese, but never in spoken, but this also did not prove to be correct. Currently I have a basic understanding of its usage, but I am still slightly confused.

Some people use it often, other people omit it entirely in sentences, what is the general rule for using it? Is "きれいよ” just as correct as "きれいだよ” besides the former being more feminine?

I have also been told that it has a grammatical importance, so that in the sentence "あの猫がきれいだと思宇” the だ cannot be omitted. Again,is there a general rule for when I should and shouldn't use だ in more complex sentences?

Thank you!