Answers

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.

The glass package and the insulation of the window are the most important. The best window would be quadriple or triple pane window with foamfilled sashes and krypton or argon gas inbetween the glass. The dense gas keeps the cold air out in the winter and the hot air out in the summer. The foam filled sashes keep the insulation like your walls. Also choose vinyl over wood for better insulation, 

the most important thing you need to make sure you have is a highly rated low-e window. low -e will reflect the solar rays away from the inside of your home so it doesn't heat up during the summer or direct sun etc and during the winter it works in the opposite manor. winter it will keep the heat inside your home keeping your house comfortable. there are many good brands of windows like Andersen, Western, Hurd, Marvin, Eagle (andersen), just to make a few. be sure and do plenty of research and get estimates from. 4-5 companies and compare dont just get estimates from 1 or 2 companies. good luck.