Can anyone tell me how to keep a house cat out of a mink box.

Dont trap around housecats.

Use a box with an entrance cover with a 2 ½ inch hole. Mink explore holes all the time and thus it's my opinion, and experience, that the hole is an attraction to them. Another advantage of the restrictive entrance hole is that when trapping a farm area, it restricts barn cats from entering as well as other non target species

If a cat wants to get your bait, a 2 1/2" hole won't stop them. Had one prove that to me once. Like others said, don't trap around an area you know or think has farm cats. Strays you have no control over.

a 2-inch hole is enough.

Ground cat makes good bait. Just saying.

If you want to know how big or small of a hole to make to exclude any animal,take a skull of that animal and measure the distance between the zygomatic arches.Then make your hole a bit smaller than that.

But dont forget to recess your trap as some animals may reach in with a foot.

How high off bottom is good for hole, does it make a difference?

Makes no difference IMO as long as they can easily reach the hole.