Yes. You would be purchasing both stock and the option from "the market."
As much as I like to joke about retiring to some farmers market to handcraft artisanal equity options, when people are talking about options on stocks (equity options) they are normally referring to exchange traded deriviatives.
Depends on the context. But in order to get OTC options you are going to need to be doing some sort of massive deal (AFAIK never done OTC derivatives trading) or it needs to be some sort of esoteric option contract. You wouldn't do this as part of a normal equity/option hedge or trade. You also would almost certainly be paying a higher price for the privilege than you would on an exchange.
Buying a "normal" option contract on an equity is almost certainly not an OTC deal and is instead exchange traded. Those exchanges are very efficient like the equities exchanges are so options are priced competitively (thanks to options traders).
Can you explain this further? Are you suggesting that you're obtaining the put option and the stock from the same source?