Answers

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. 

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.  

Check the weepholes and make sure they are not clogged. These holes allow water to drain outside of the window, and are on both sides of the window. If they are clogged then the water will drain wherever it can - inside your walls and house. When the window is closed, you are making it harder for the water to get inside. I suggest taking the tip of a knife and just cleaning them out to see if that helps. 

Is there a leak when it's raining or is it just when the window is open in general? If it's simply when the window is open, then there could be several varying possible reasons as to why the basement leaks, solely when the window remains open. Bad timing, as off putting as that sounds. If it's when the window is open during a rain fall, the window could have lost its seal. Rain could leak in through the wall and then to the basement ceiling. If that's the case, I would recommend a new window seal, or replacing the entire window altogether. If you have further questions or concerns, please feel free to call our office. 403-343-0393. Thank you!