There are two kinds of radiation in the Metro series: direct radiation (gamma rays) and areas with radiated dust what kill you only if it reaches your lung. The mask protects against only the latter (in the game and in real life).
Walking in an area with radiated dust Artyom's Geiger counter will be in the middle yellow range and "ping" slowly. Artyom will start to suffocate if not wearing a mask. On the other hand, when exposed to direct radiation the Geiger counter will be in the red section and "ping" quickly, in these areas regardless of the mask Artyom will die in a short amount of time. (It is possible to spend more time in these areas while constantly healing yourself.)
Does damage to Artyom's gas mask affect how much damage he takes from the environment or how quickly his gas mask filter needs to be replaced?
The game mechanic differs between the original, Redux games and Exodus. In the original and Redux games, you can crack your mask which will lead to faster usage of air filters, however, if your mask got broken you would suffocate in a short period of time if no new mask is found. (Extra masks were present game, in Exodus the mask repair mechanic replaced it.)
In Exodus, your mask has two states: working or broken. (Technically there is a cracked state but it works the same as it would be pristine.) When your mask brokes you would start to suffocate and need to repair it on the field with duct tape using your inventory.
Does wearing the gas mask provide increased protection from physical damage?
No, it does not, but it exposes the gas mask to possible damage (when a monster hits you) so fighting while being in radiated areas is not advised or with extreme caution.
Footnote: The mechanics are briefly explained on the Autumn level when Tokarev briefly mentions that the protective gear will save you from the radioactive dust but not from the direct radiation.