There was about 3 cups worth of water total but I was concerned that the moisture would cause mold so I drill several holes in the ceiling to let air get into the cavities.

Is there moisture coming from the cracks, Is the house level, What year was the house built, Location is a big factor.

Most recently (summer of 2015) I was putting in shower high hats (recessed lighting), but only was required to use a "shower trim" It seals the high hat off from the moisture of the shower.

Will cost significantly less in labor and a roofer would most likely come out and felt it that day as to avoid moisture damage until they could get out and install the shingles.If the damage is next to a dormer or peak and the flashing isn't there or was installed incorrectly, you may be able to call and get them out there.  

Likely moisture and/or rain has been getting through.Mold: General climate in the area is dry, but we get enough major rain showers in the spring and fall that any exterior holes likely allowed water to get inside.Bank offer: Do you think I can actually get the house with an offer at 50K.

The problem with particle (pressed) wood is it is porous and any visible mold treatment may not penetrate deep enough.

@Donny Dormady Considering the fact that you live in Alaska and the entryway will be exposed to moisture, I would go with a good commercial-grade vinyl plank as @Nathan G. suggested.

Less risk of mold, mildew, moisture build up.  

My theory is maybe an installation or moisture issue, but I really have no idea.

I will have paid for all of it as the repair and getting rid of water/moisture was an asap thing so the tenants could go back to their lives.