Could an immortal, invincible, unstoppable man, who is weilding a gun with unlimited ammo, theoretically wipe out all human life on Earth?

Ah - one man? I think by the time the guy killed everyone on earth, which would take some time, new births would have re-populated it. The math is daunting - figure one minute to kill each person (including travel to go to the next one, etc., on average) so that's 60 per hour, say 1,000 per day (rough estimates, make the math easier) 350,000 per year, say 7 billion people in the world, so that's 20,000 years just to kill everyone once.

No, he couldn't live long enough even if all 7-plus billion people lined up one by one and didn't need to be hunted down. Do the math dumbass, and try to think happier thoughts.

Depends on his rate of fire. There are 353,000 kids born per day (245 per minute, 4 per second). Assuming that the man is human other than the properties you've assigned him, it would be difficult to find and kill at that rate throughout the world with a gun. If he selectively killed the childbearing population, or only one sex, so that reproduction was impeded, he could do so with a lower rate of fire (but it would still be measured in shots per second). But this would take a great deal of cooperation and resources of the very population he is trying to eliminate.

No. There are too many of us.

Like painting the forth bridge

Unless you define his limitations more clearly, it is impossible to say. If he can't breach armour plating, huge metal doors, has superhuman strength, speed or is able to fly, then there are any number of ways to incapacitate him, or remove him as a threat.