Keeping an eye on someone means you are watching them. You could be watching them for their own safety, or you may find them suspicious.
Example: "I wanted to go shopping, but Mom has me busy keeping an eye on my 2-year-old brother instead."
Example: "Even though it was Halloween, I decided to keep an eye on the shopper with the ski mask, just in case they were a real robber trying to blend in with the people in costume."
jump a lots
This is not used as much as it once was, but people will know what you mean if you use it; it's not obsolete.
The image is of you pointing an eye at something or someone. It means to watch carefully.
If something "catches" your eye, it has snagged your attention. You are focused on that thing for whatever reason.
The idiom "eagle eye" refers to having an acute eyesight, or an intently watchful eye. It refers to the sharp eyesight of an eagle, which are known for seeing fish from hundreds of feet up.
Nothing. I'm afraid you've gotten the idiom incorrectly. It should be "the apple of her father's eye" and it means she is his special favorite. The apple of the eye is another term for the pupil or center of the eye.
"Don't fool around because I've got my eyes on you."*The idiom for "watching someone" is often just the singular "eye" (e.g. "I'm keeping my eye on you.")
It means you have one eye open. If you mean the idiom "sleep with one eye open," it means that you are staying alert at all times in case of disaster.
It means to agree / be in agreement about something. Often it is used in the negative form eg They don't see eye to eye and the problem is not getting solved.
The idiom "keep an eye out for" refers to watching for something or someone. An example of a sentence using the idiom would be: Jeff should be arriving soon, so keep an eye out for him.
No, the idiom is "turned a blind eye" as in "she turned a blind eye to his suffering." You don't change an idiom around or you lose the meaning.
It's not an idiom because you can figure out the meaning by context. If something pleases your eye, it's nice to look at.