Did George do the right thing by shooting Lennie? Why or Why not?

Yes, George's mercy killing of Lennie neatly parallels the events of Chapter Three, when Candy allowed Carlson to shoot his malodorous old dog. Steinbeck is even careful to involve the same Luger in each killing. Whereas the meek and passive Candy proved unable to do the job himself, George shows no such weakness. As has been proven beyond a reasonable doubt at this point, Lennie's lethal innocence is not compatible with the world. He cannot learn to change his ways - he cannot even understand why the "bad things" he has done are bad. The fate he would meet at Curley's (mutilated) hands - likely a drawn-out, vengeful lynching - is enough to convince George that his only real option is to make Lennie's death as quick and painless as possible.