Answers

The difference between new construction and replacement is how much of the existing material they remove/replace, and how the windows are secured to the opening.

In new construction, they remove everything on both sides of the wall, interior and exterior. In replacement, they only remove the inside trim and the window box. I think I have that right? New construction allows you to really look at both the interior and exterior of the opening and make sure you don't have dry rot, make corrections to square/plumb/etc, and nail directly into whatever all that stud/framework stuff is called, but can cause minor/cosmetic damage to the exterior, esp to stucco. Replacement is just removing the interior trim, popping out the old window box and popping in the new one. In my case, they even used the old trim, just nailed it back up and caulked around it.

I had my windows popped out and replaced, primarily because I'm in a 65yo stucco rancher in an area that did a lot of building with redwood. Conventional wisdom says that is exactly the kind of house you should do a new construction installation, rip everything out and have a good look around, but I know I'm only going to be here until my kids graduate from college and when I sell the house it will be razed for a McMansion, like every other house on my block. I just need to get another 10 years out of it, I don't want to put a single $ in I don't have to.

Two things I did not anticipate:

One, my installation did NOT include any touch-up type work, I was responsible for any cosmetic touch-up repair and repainting of interior wall and trim affected by the replacement.

Two, when you put dual-pane acrylic windows where there used to be single-pane wood, you lose nearly all of the interior casing/frame/jamb, whatever you call that part that you would use for an "inside mount" of any kind of shade. I didn't realize I wouldn't be able to use any of the pleated, honeycomb or black-out shades I'd just ordered...

The window (frame) surface is cold, but the air in the apartment is humid. Humid air condenses on a cold surface (when it drops below the dewpoint temperature).

So you can either make the window frame surface warmer, or you can make the air less humid.

Another approach to making the air less humid is to run the air-conditioner, but that makes the house very cold. The dehumidifier usually makes the air warmer, so it's probably preferred, although way less capacity. Some fancy houses have dedicated whole house dehumidifiers (but maybe they also have better windows!).

Window films (the type that shrink with a hair-dryer) attached to the window trim/casing can keep the humid inside air away from the cold surfaces. Added bonus of keeping cold air leaking through windows out of the house.

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.