ok Teod...you have almost answered you q...As you have said mewtwo was in a lab.
Some scietists collected ancient DNA from Mew...then they tried to creat a clone of Mew....they created a different Pokemon that it was too strong to be kept in a laboratory.
Jiovanni,the team Rocket's boss discovered the existance of Mewtwo and captured it ,to use for a bad cause of course.
As for the games FireRed and L eafGreen ,GameFreak just wanted to use the legend of the 2 Pokemon and they created the cinabar's mansion for the games.
FARAWAY island is the place that that mew lives.....its just a part of the game,u know rayquaza lives in sky pillar ,kyuogre in Marine cave.etc.
The anime and movies have nothing to do with gemes .....they are 2 difffernt categories.
P.S;it would really help if you could watch the 1st Pokemon movie.