I love worldbuilding questions which touch biology!
I would argue that it's not so bad as some other people here present. The main question is, where does this power come from? Evolution? Magic? Technology?
Depending on the answer to this question, some the problems listed above could be easily or very easily solved.
1. Saccades and blinking are controlled by nervous system and are thus easiest to solve - in fact, nature and engineering has already solved a number of similar problems.
ex 1. WWI fighter planes had initially a huge problem with firing on one another because the machine gun was mounted behind the propeller, so when the gun fired, the bullets often hit the propeller fans. This was solved quicklly by a simple mechanism which synchronised the firing and the propeller. The same could apply here, with a switch blocking blinking and saccade reflexes during the beam firing.
ex. 2. Swallowing vs breathing - You don't have to control Your breathing while drinking a tea, dont You? The nervous system is wired to do that for You.
The only question is, how was the nervous system taught to block the reflexes during laser fire. 693
Evolution could do that for You easily, otherwise it was directly magical or technological manipulation (hard stuff).
1. Burning the lenses - well, we have two options here. First, who said that the laser has to originate from the eyeballs themselves? Maybe it's origin is right in front of the eyeball? Second, there are a lot of transparent materials which have no problem with passing the laser through - especially if the laser is really, really monochromatic. Again evolution would have no problem doing that, since it already produced lenses transparent in the visible spectrum (and the wider the spectrum, it is probably harder to adapt to it). Magic could probably do the same, the technological solution would probably be the hardest here.
2. Scattering - practice, practice, practice. Again, evolution could help a lot by doing some additional hardwiring in Your brain, and so could technology and magic. Frogs are super precise with shooting their tongues, and so is <PERSON> in Trigun (well, he is precise with guns, not tongue).
3. Power supply - this depends on the physics of Your world, but since we are discussing laser-shooting-eyes, I assume that it's at least a bit different from ours.
I also wanted to point out that whatever <PERSON> in X-Men is shooting, it's not laser, it carries way too much kinetic energy. 477