You wouldn't be able to hold an oxygen mask on your face with the nearly 15 PSI differential. Surviving there without a suit would be like surviving in a balloon lofted to about 30 km altitude -- not feasible. If somehow you did, exposed skin would rapidly freeze.
Yep. The thing that matters for
respiration is oxygen partial pressure. On the Earth's surface, your body gets about 2.9 psi of oxygen mixed with nitrogen and whatnot. If you're climbing Everest with an oxygen mask, you can breathe yourself about 2.9 psi of oxygen almost all by itself---with only 1 psi or so of nitrogen diluting it---so in principle you can maintain sea-level-like respiration. Reinhold Messner, climbing Everest without a gas mask, is surviving---probably just barely---on 0.8 psi of oxygen (20% of a 300 mbar atmosphere). So perhaps we can call that a lower limit.
Mars' surface pressure is about 0.1 psi, so if you were holding an
ambient pressure gas mask (like scuba gear) feeding you pure oxygen, you'd be a factor of 8 short of the Everest-without-gas lower limit.
Suppose you could crank the gas mask pressure up to 0.8psi and have it (lightly) strapped onto your face. There, now you're getting as much pure oxygen as Reinhold Messner. With this setup, without a pressure suit, your lungs are holding air at a somewhat higher pressure than the outside environment---this means they want to expand, and it's hard work to exhale. Every breath you take would feel like you're blowing bubbles through a hose whose end is 1.5 feet underwater. Try it! It's not going to make your lungs pop, but it's a good amount of work on every breath.
So the answer, I think, is MAYBE JUST BARELY. If you had Reinhold Messner's red-blood-cell-count, and you're able to do a good amount of work per breath, and you have an 0.8 psi overpressure oxygen mask---yes, you'll be able to breathe on Mars without a pressure suit.
I bet that a very low-grade pressure suit----a handful of rubber straps around the chest, at the level that'd make it difficult to inhale on Earth---would take a lot of the load off of breathing. Maybe you could get up to 1.0 or 1.2 psi.