If you want to be playful, you can definitely use “gone missing” … like, if you have a small child and the child is hiding under a blanket, you can say, “Huy has gone missing somewhere in the house” and that would be a playful use of the term – exaggerating to be a little funny and playful. Same goes with a cat

Thank you for editing, as I thought about this sentence the last part I wanted to joke about for fun. I rewrote: My friend is looking for the cat that has gone missing somewhere in the neighborhood.

I would agree with this except for the part that says “in the house.”

“Go missing” is used when it’s hard to find something. It’s not hard to find an animal in a house (usually), so “go missing” is better for pets or animals that are outside. So you might say, “….looking for the cat that has gone missing somewhere in the neighborhood”

Exactly! Some aren’t found until well into mid-day Saturday…

Great example of “go missing.” A few tips for you. “Take care of our children” is a broad term that means, to meet all of their needs. In this case, I think you mean, “to protect them.” In this case, I think the best term would be “keep a close eye on” which means, to watch something carefully in protective way. “All the time” is okay here, but you can also say “at all times.” And finally, we say, “in the blink of an eye.” So:

“In crowded places, we should keep a close eye on our children at all times because they can go missing in the blink of an eye.”

This is correct and a good situation to use “went missing.” “Before or after” sounds just a little not quite right, almost like the reader is supposed to guess. But I know what you mean. You might say, “My luggage went missing sometime after I checked in”; or even, “I’m not sure whether my luggage went missing before or after the flight, but it’s not here.”