Okay well, first of all, it's already known that the Anime and Game timelines take place in two different parallel universes (and the Adventures Manga also has its own universe, which of course gives the other, somewhat less popular and important, Pokemon Manga their own universes too.)
There are multiple legendary birds. There have been several legendary birds in the anime (and they've all been different individuals). They of course got featured in Pokemon Movie 2000, along with Lugia, who would later reappear as a baby Lugia (and of course, suggesting another Lugia which is possibly the same one from the movie. However, the movies aren't canon to the anime, so it's not likely)
Later in the Diamond and Pearl series Team Rocket saw a Zapdos fly by. But if I remember correctly, they could not pursue it because it shot down their balloon.
Articuno appeared in the Johto League Saga. I can't think of a non movie appearance of Moltres, but I'm pretty sure there has been one.
The Legendary Beasts have shown up a LOOOOT in the anime. In the chronicles, in the Pikachu shorts, in the Johto Saga, in the third movie, in the 13th movie (but they were illusions. however, I think the ones in the third movie were also illusions), etc...
Red is the Ash of the game world; Ash is the Red of the anime world.
The Origins also has its own universe. If, perhaps, there were another Red and Blue (oh and of course, Gary is the anime version of Blue)remake, it would probably be based on the Origins Special.
Ash has never met Red, and never will (unless the anime writers decide to go Super Ultra Weird Mode...although this would be possible considering how a few Pokemon can cross through dimensions).
You see, if they were in the same universe, this would not make sense. In the anime a guy named Dr Fuji created Mewtwo with Team Rocket. In the games, he was seemingly only made by TR (although Fuji appears as Mr. Fuji in the games, and this is even further hinted at in the Origins Special). Later of course, another Mewtwo was cloned in the anime.
Aaaannyway! Sorry if that was too long. But the fact is: the games and anime are in different universes, and there isn't a confirmed number of Legendary Pokemon that can exist in any universe. I believe that after the heros (players) of the games catch Legendary Pokemon that are very important to the universe, Arceus (if he was captured then.. I have no idea :O), or another important Pokemon creates a replacement for them.