The nit-picking answer to your question is yes, and furthermore, neither American nor English writers put a hyphen in a stock phrase like like new when that phrase is unambiguous, as it always is when it is a predicate adjective. The working answer to your question is yes, and drop the hyphen, and while you're at it, drop the quotation marks. Like new is perfectly familiar diction, which means that it doesn't require typographical emphasis. Quotation marks are used to dr aw attention to a new coinage or an old word used in a new way. In this case, the quotation marks are likely to make your readers think that you are lying about the condition of the floors, and that they are not like new at all, but rather that the owner simply wishes them to be perceived as like new.
Question marks are a little more complicated.
They are placed inside of quotation marks when they are part of the quoted matter. Problems arise when the question mark attaches not to the material inside the quotation marks, but rather to the frame of the sentence. In this example, the question mark belongs to the why clause, not to the quotation. Writers who feel the need of a comma are seldom distracted by prescriptive grammatical rules. I recommend grammatical psychoanalysis.