Not really. The 'wear layer' is the top layer. If you've already worn down through the paint all the way to the primer then its 'hardiness' is rather immaterial, wouldn't you say?
The purpose of primer is to even everything out and provide the 'key' for the other paints to stick to. Primer is your friend. Yes your door needs priming.
Yes indeed, also your first coats will probably raise the grain as it wets the wood, if you keep painting over these imperfections they just get bigger and bigger as the coats build up.
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