well, with those alternatives, its kinda obvious. suicide, hires aother to shave him and both.
but for the traveling back in time and killing yourself. that would work itself out seeing as if you went back in time to kill yourself. you would die, then you couldn go back in time to kill yourself seeing as you are dead. so you would live on. if we then go by the theoy that time is constant, yay endless loop, you have destroyed time. gg. but lets say its in flux. then at one or another repetition of events you would decide not to go back in time to kill yourself. like, why would you do such a thing.
as for the barber. I have no clue why anyone would count that as a paradox at all. same goes for the day night thing.
"A Mars Colonist says 'All Mars Colonists are liars'." this one mars colonist could still be a liar even if not all are. not a paradox.