Is there any way to check the enemy's firing range or Overwatch range?

Overwatch reacts to anything moving in the unit's line of sight.

Alien sight range is the same as yours. If you can see them, they can see you (unless you have elevation).

No, there is no explicit indication of whether a specific alien can hit a specific target.

Range of most weapons is equal to sight radius, which is somewhere between 17 and 18 grid cells.

• If the alien and the target are somewhat aligned to grid lines, you can count cells.
• If the angle is funny, you can try to measure sight radius - say, save the game and try to remove other soldiers who see the same area.
• If a target in question can shoot and has a normal weapon (not a sniper rifle) you can check - if you can hit the alien, it can hit you.

My personal favorite way of dealing with such situations - put myself into shoes of a scared human on a battlefield, take a wild guess and make a run for it! :-)