There is a lot going on in this. The biggest problem is the fact that the structure of the example sentence you found is a bit more advanced than you are used to. The example sentence is right. If you were to break it down
食べている = currently eating
人は = "a person" [the は marks the "person" the focus of the question]
います = exists
か = ?
誰でも is the closest thing to "anyone" in Japanese, but it is not used here because it is implied by the use of the word person. The example sentence could also be translated "Is someone eating?" rather than "Is anyone eating?"
To use an easier sentence structure I think you would want "誰かが食べていますか？" which is (again) closer to "is there someone eating?" than "is anyone eating？" but using ”誰でも” here sounds very awkward.
An important point to mention is that when you use "誰でも" the でも replaces the particle you would normally use, so if you were to use 誰でも you would remove the が entirely.
Xはいます is confusing in the example you've included because of how advanced a sentence structure the example you found is. (I don't think you would be taught sentences like that until you reached Japanese 201.) X is not "any" it is merely the topic of the sentence, as it would be in any structure where it was followed by は.
In all of your example sentences you want to look at the います which in all cases indicates existence. You would translate
います ＝ "there is"
は ＝ "a"
猫/車/花 ＝ "cat/car/flower"
EDITED: However, as Yadokari pointed out, since います can only be used for living creatures you would usually use があります for the car or flower, leaving 猫はいます the only correct example. Depending on the sentence all 3 would more commonly be written with が than は, but that gets into an entirely different question.
I am sorry this got so long but I hope it can help you a bit!