Answers

Looks like this may have been someone's remedy for a deteriorated / rotten mud sill (the flat wooden plate which sits atop the foundation wall, upon which the joists bear).  If that is the case, it is possible that you might be able to remove the latter obstruction if one can restore the proper / original condition over the window.  That would want to be examined and undertaken by a professional.

As with any such thing, it is difficult to say for sure without a full on-site examination by a home imporvement professional with structural knowledge and experience in these types of repairs.

You can more than likely call the manufacturer that made the window and just have a replacement sash made.

Its sounds like you have a double hung window that slides up and down check to see if the outer top window can slide up then lock it. If the glass is sperating from the frame you can take it to a glass replacement company and they will fix the frame and glass if not vinyl welded

I would recommend replacing the window.

sealing or adding any aftermarket seals would most likely sacrifice operation 

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.