There are a couple of mistakes here:
For one, you want to be consistent with the politeness level. For example, the first sentence is in plain form, while the others are in polite form.
The second sentence is using a general counter for Penguins, which is the incorrect counter. In this case, you’d use the 羽 counter, which is used for small birds and rabbits (I know, lol. Blame the monks for that one). The verb usage is strange too, so I replaced them.
Lastly, in the third sentence, you’re using 彼 as the pronoun, but I think Japanese people either use (name of pet) + ちゃん, or the sound the animal makes + ちゃん. When the name is not known, I believe they simply say (animal) + さん. I’m not 100% on this, since it might sound a little childish, but that’s the pattern I’ve seen. Perhaps you can just stick to ペンギン by itself.
Anyway, the end result would be something like the following:
ペンギンを見るために南極にヘリコプターで行きました。それからイグルに入ってペットとして一羽を取って家に帰りました。今ペンギンはフロゼンの野菜とフリーザーに住んでいます。
HTH!