Answers

That is Oriented Strand Board (OSB), typically used for sheathing.  Looks like it has been used exposed to weather, which is not its intended use.  There is no reasonable repair for that, other than replacement... with materials suitable for whatever the application might be.  Call a professional contractor.

I agree with what Yuri said and would like to add that it is imporatant to have it removed properly so as not to cross contaminate. Mold spores are microscopic in size and easily dispersed if not handled correctly.

The leak and mold are time sensitive. Mold can be hazardous to your health. Dealing with mold properly means removing the sheetock and insulation once the leak is fixed. Don't let anyone tell you they can seal it in place and paint ove it. It must be removed. The foundation work can get vey expensive if you wait too long so don't put that off too long.

Hope this helps.

It can definitely be repaired, it would need extensive spackling, most likely with 2 part Bondo, then sended primed and painted.

this is something that should not take more than 6-8 hours in total, obviously it does require some skill.

The picture shows the door is stained wood.  The wood is repairable but the door will have to be painted or the repairs will be visible.

Do you know if the estimate from Home Depot was to replace the entire door assembly or just the (door(s)?).   If they estimated the cost including removal and replacement of door jambs, threshold, and door trim, then an alternative that could cost less would be to replace the door slabs only and keep the existing jambs and trim.

Cut the screw if the other methods fail.

Its painted; go to hardware store.  Buy spackle and sand paper.  Sand it down, spacke the scrapes.  Sand the spackle.  Repeat again, though after this spackle prime the door.  Then spackle and prime and paint.  You are looking at spackle 3 or more times, priming it more than once and then paitning two coats.  Alot of work.  But less expensive than a new door.  

Try securing the anchors with ViseGrip pliers, but be sure to only secure the anchor, not touch the screw.