Is this verse from Lady Gaga grammatically correct? [duplicate]

You're right that it depends on the rest of the phrase (subject or object):

• "My wife and I are eating an apple" is correct because "my wife and I" is the subject of the sentence. You could replace "my wife and I" with "we".
• "You can see my wife and me in this picture" is correct because "my wife and me" is the object of the sentence. You could replace "my wife and me" with "us".

You can often check this quickly by removing the other person from the sentence and see if it makes sense. For example:

Subject:

• "You and I like apples". Removing "you and" gives "I like apples", which is correct.
• "You and me like apples". Removing "you and" gives "me like apples", which is incorrect.

Object:

• "John gave the apple to you and me". Removing "you and" gives "John gave the apple to me", which is correct.
• "John gave the apple to you and I". Removing "you and" gives "John gave the apple to I", which is incorrect.