Unless the top of the well is so low that surface water gets in, like in heavy rain flooding into the well, the cover presents no problem so long as it keeps out mice and squirrels. In fact, covers must admit air (inhale and exhale) so that the pump can draw water from the well and and more water can flow in from the veins below- the air volume of the well must change accordingly. Coliform might not necessarily make you sick. I suspect that I have been immune for many years. E-coli is another matter and you do not have that.
Arsenic? Can't offer advice about that. But some 60 years ago, feed supplements for chickens contained low levels of arsenic which was believed to be beneficial for them. Seek additional advice about the arsenic.

Thanks, bus driver! We shocked the system and flushed it out this week. We drew another sample this morning. If it still tests present we will start looking at the pipes and work our way back to the well itself to see if we can figure out where it's getting in.

My understanding is insects can also cause well water to test positive for total coliform. You might need to check to see if insects are contaminating the well.

Good luck with the second test. We also have total coliform, have shocked the well twice and will have our third testing next week.