"The eagle eats a mouse."
Translation:Ο αετός τρώει ένα ποντίκι.
Not really, the meaning changes a bit. Without the article it would be more like "The eagle eats mice", as if a mouse is part of the eagle's diet, the same way someone may ask "Τρως αρνί;" = "Do you eat lamb?" Answering yes does not mean that you're eating a lamb, but you will eat it as food. The lesson sentence is more specific in its meaning than that.
Is ποντίκι related to οίκος?
Not at all.
ποντίκι is a diminutive of ποντικός, which comes from Ποντικός μυς "Pontian mouse" (see https://en.wikipedia.org/wiki/Pontus_(region) ).
So, a little bit like how Latin iecur "liver" was used in the combination iecur ficatum "fig-stuffed liver" (a favourite dish), and then the adjective ended up being used as the word for "liver" in descendant languages, e.g. Spanish hígado, French foie, etc.