There are dozens of different Regi locked up in remote locations, waiting to be awakened. However, there is only one (maybe two) Regigigas. -> not right, there are 1 of each.
There is perhaps one hundred, maybe two hundred Shaymin. They're more endangered then Farfetch'd is, though. -> also only 1
Obviously there's only one Arceus. -> right
There was herds of Latias and Latios, but their population dwindled to the hundreds. -> in the 5th movie, they have a second latias, that was the father of the on we know, so they can make kids.
There's perhaps less then ten of Kyogre and Groudon (with one main Groudon and Kyogre) still living today, and Rayquaza may be the last of it's kind, or there are two Rayquaza. -> 1 of each of them
Dialga, Palkia, and Giratina are each unique, with the exception of a baby of one of the three, made by Arceus for the hero of HG/SS, perhaps for in the case that something might happen to said member of the trio. -> WTF??? Dialga, Palkia & Giratina are unique, so no babys. Or there must be another dimension with a own Time, Space & Reverse.
They only made one Mewtwo. -> indeed
Mew is critically endangered. -> also 1
Manaphy is in the hundreds. -> not right, they can make a baby (phione), but there are not a great amount of them...
Lake trio are unique, as far as canon is concerned. -> indeed
Celebi has many copies of itself, but all in all, it's the same individual. There are only two true seperate celebi (or maybe there's a bajilion of them, I don't know), a green one and a pink one. -> There is only 1 celebi. In the 4th movie can you see more of them in the end, but that are the same one, from another Time (because Celebi can go in time).
The legendary birds can mate, and have a small population. -> 1 of each
Heatran are perhaps one of the more "common" legendaries, dwelling deep underground where no human could survive going down. -> only 1
Legendary beasts (Suicune is NOT a cat, and Raikou is NOT a dog!) can go either way, perhaps the trio we see are merly the dominant members of their race, or perhaps they're the only members of their race, who knows? -> they are each unique, and they are dogs, because in G/S/C say a guy in the temple anything from the 3 legendary DOGS!
Jirachi is unique. Or maybe there's one sealed away in another cave somewhere. I dunno. -> only 1
Thre are currently four known Deoxys, one for each form resepectivly, but there may be more roaming around the Pokémon world, crash landing in meteorites in remote locations. -> there is only 1 Deoxys but he can change hes form.