Does vegetarians eat eggs and milk and cheese?

after answering that question heres a follow up: If so isnt that also eating them? I mean you just eat the eggs but they would have been chicken? and the milk is part of the cow? arent they fed when they are little? doesnt that mean that they also eat their mother? (sorry about the word but I dont how else to say it) Also they say beans can produce gastric acids and is not good for old people cause it developed arthritis, and even when youre young you would just storing up those acids into your body., so what is there left to eat for the vegetarian cause protein and other nutrients are very essential for ones health? please answer all the question if you can and not just selectively and please with sense thank you.