1) Le chat est à l’arrière de la maison = The cat is at the back of
the house (like opposite to the front)

Yes, correct.

2) Le chat est au fond de la maison = the cat is right at the back
(like in the depths of the house)?

That one I would think it mean the cat is hidden in the basement, or somewhere depth inside the house (like a chimney, etc..), but it’s open-ended on the location inside the house.

3) Le chat est derrière la maison = The cat is behind the house (but
doesn’t specify particularly)

Yes, correct.

4) Le chat est en arrière de la maison = The cat is just behind the
house. I reached that conclusion as I understand that “en arrière de”
can mean “in the background of”?

Yes, correct. It mean the same thing as derrière la maison. So can be in the backward, etc.. , but somewhere behind the house.

I’m trying to understand the difference between these things. Do I understand correctly if I say:

1) Le chat est à l’arrière de la maison = The cat is at the back of the house (like opposite to the front)

yes

2) Le chat est au fond de la maison = the cat is right at the back (like in the depths of the house)?

yes, at the very back of the house

3) Le chat est derrière la maison = The cat is behind the house (but doesn’t specify particularly)

yes, the cat is behind the house somewhere

4) Le chat est en arrière de la maison = The cat is just behind the house. I reached that conclusion as I understand that “en arrière de” can mean “in the background of”?

yes, the same as derrière la maison … in the background of doesn’t really apply here.

🙂