"A mouse is an animal."
Translation:עכבר זה חיה.
I think a useful way to make sense of the complicated rules about the copula is to be aware that it is mainly used to separate two words in order that they are not read as a unit. דּוֹלַר־כֶּ֫סֶף is a silver dollar, so you have to say דּוֹלָר זֶה כֶּ֫סֶף a dollar is money, or נֶ֫שֶׁר יָפֶה is a beautiful vulture, so you have to say נֶ֫שֶׁר הוּא יָפֶה a vulture is beautiful, but הַנֶּ֫שֶׁר יָפֶה can only mean the vulture is beautiful, because you can not combine these two words to one syntagma.
I would think the use of זֶה makes it more of a definition, like "Can you define, what a mouse is? Yes, a mouse, that is an animal". On the other hand הוּא is a simple statement, like a mouse is brown or a mouse has whiskers, whiskers and brownness do not define a mouse.
Well, this is true of declarative sentences (i.e. simply statements that relay information) like הַבַּ֫יִת שֶׁלְּךָ זֹאת דֻּגְמָא טוֹבָה your house is a good example (where בַּ֫יִת is masculine, but זֹאת agrees with the feminine דֻּגְמָא). In definitional sentences like a X is (by definition) a Y, זֶה tends to be uninflected: הֵרָיוֹן יֶה לֹא מַחֲלָה pregnancy is not a disease or דִּבּוּרִים זֶה לֹא מַעֲשׂוֹת words are not deeds. The copula is a notoriously difficult topic in Hebrew, sorry.