First, are you sure it’s 3/8"?? I’m not sure you can sand one that thin.
From personal experience, I heartily recommend doing as much work as possible before you move in. Living around remodeling is hell-on-earth. Floors especially - they’ll turn out much better done all in one go, but it’s difficult to not walk on floors when you live there. 
I think you could probably scrape up the old pad. You might try to think of something larger than an ice-scraper though. :eek: There’s a good chance that the floor underneath will need some work, but not necessarily heavy sanding.
If the floor is in good shape, I’d consider doing a strip and refinish - there’s not nearly as much concern about accidental damage. You can strip a floor just like you do furniture (i.e., stripping chemicals and lots of elbow-grease). It won’t take out all the signs of age and wear, but that’s part of the charm IMO. First you’d need to figure out what the current finish is, to know how to best remove it. At any rate, you’d only sand as much as you would for any other refinishing. An orbital sander would work great for this, I’d think.
If the floor is really rough, it needs to be sanded. Sanding will take a fair bit off the top and remove rough spots, many stains, etc. As you noted, sometimes a stain or something just won’t come out. Give up and call it ‘character’, or you’ll wreck the floor.
DIY floor refinish is a hellacious amount of work, and you can truly muck your floor with the sander if you’re not careful. Also, the sanding machines you rent are often in less-than-optimal condition, making it even easier to divot the floor.
The finish part is definitely DIY territory, if you’re fairly handy. A lot of work, but feasible.
I’d check with the pros first. Tell them you’re overbudget. Ask about stripping rather than sanding. They will want to sand - it’s a lot quicker and more cost-effective for them - but they should be able to tell you what the finish is, how hard it would be to remove, etc. Ask about orbital vs. drum, how many passes it would need, etc. (This should give you a good idea of how much sanding the floor needs.) A good contractor will work with you - the contractor we hired to sand floors gave us all kinds of advice on doing the finish work. Once you’ve gotten a few professional opinions, you’ll have a much better idea of which way to go.
The first floors I worked on, we hired just the sanding done. We did all the rest of it ourselves. That included staining the floor in one room (too many dark stains to leave natural) plus three coats of sealer (kitchen floor* - I wanted that puppy SEALED), then no stain and two coats of poly on the rest of the house. Probably about 1000 sq ft total.
You’re talking hands-and-knees to put down the polyurethane, attempting to keep it even, without brush strokes, etc. Then ditto to hand-sand the finish. Then again to vacuum up every speck of dust. Repeat for as many coats as you can take. It didn’t help that we were working in a house with no central HVAC, but had to keep the windows closed to keep (most) of the bugs and dirt out of the floor. (Don’t do that - we were young and stupid. I’m surprised none of us passed out.)
We’ve also done the entire upstairs of the new house (approx 1000 sq ft) plus a friend’s living room (appx 400 sq ft) in the last few years, including the sanding. It can be done if you’re very, very careful. We are fairly handy types (one professional remodeler - who does NOT do the floors in his remodels, one experienced amateur remodeler, plus a couple of us who are fairly familiar with tools).
So get some professional opinions, then decide how risky it would be to do it yourself, then decide if the cost + risk is worth it.
Good luck!
*I had more arguments with people over a wood kitchen floor. The floor was in great shape, just stained, but everyone and I mean EVERYONE I knew insisted that you just COULDN’T have a wood floor in the kitchen. Nowadays, twenty years later, everyone is doing it. Grrr.