If you have a free game with paid features, it's up to you to decide how you can give it for free to specific users. Maybe ask yourself an e-mail when buying or something similar and check if this e-mail is entitled to a free feature.
If you want to offer a paid game for free to specific users, the only way I can think is to use Store-specific features. For example, on Play Store, you can allow specific emails to get your game for free, on Steam, you can generate keys that you can distribute, and so on.
Use the built-in features of Play Store and iTunes Connect. This is not an option for me as they limit the number of coupons I can create.
I'm not sure what you mean by that. I've only used the Play Store and Steam in the past, but from what I remember, the limit is high enough not to really matter. Unless you plan to give the game free to a large group of people, in this case it becomes tricky.