I think remakes should be made as soon as the gen that is being remade isn't playable on the handheld (Although I'd love to see gen 4 being remade as soon as possible of course). But I've heard other people say that this isn't true, and that gamefreak goes by fan demand. Has gamefreak said anything about this? If not, what do you think?
No, they go in order, Gen 1, then Gen 2, etc.
But as for your first question, the answer is no because HeartGold & SoulSilver were released when Gold & Silver were still playable on Nintendo DS Lite.
I think that remakes are made in order by the gen. Gold and Crystal were remade into HeartGold&SoulSliver. That was Gen 2. Next was Ruby/Sapphire/Emrald which is Gen 3. Their remakes were made known as Omega Ruby&Alpha Sapphire. All remakes have been in order by what generation it was in. So if remakes were to be remade again, Gen 4 consiting of Diamomd/Pearl/Platium would most likely be next.
I genuinly believe that the first new remake we can expect is the 1st Gen games. (or a remake of the remake). It's the only generation currently not available on DS. It gained a load of popularity with Pokémon Origins, and the 4th Gen games are still very much playable on current handhelds, so I doubt we'll get a remake of those any time soon.