Men's watch, it has an alarm function. Not much you can do to improve the dial condition other than repaint it which diminishes its value.
I'd say the current value restored around $1200-1300.

Check here for approximate costs to do that:
https://www.classicwatch.com/vintage-watch-repair/

The movement is caliber K480/CW dating to about 1959.

I don't know if it has a model name. But from the photos, the case reference number is 7837. Dates to about 1952.

Zaf

Very tricky valuation on that watch. You rarely see them. I think if it's "cohesive" i.e the military marking are legit and the movement serial matches those markings then it's worth quite a bit. They've been hard to authenticate as I've never seen an A-15 spec. Just haven't had one in ages to give ...

If I am interpreting the colors correctly, it's a stainless steel back. If that is the case, the top is usually made of a lower Karat gold, like 10k or 14k.

Can you provide a photo of the back? It would tell me what the case is made of.

These are made by Jaeger-LeCoultre and have a similar valuation.

They won't have anything about that watch's history beyond what I told you as they have no record of what was done to USA watches. They only shipped the movements out of JLC CH, the case was made in the U.S.A. JLC CH was no records other than that of the movement.

Save your $.

Cal 480 is decent.

It's a caliber K480/CW, dates to about 1963.

If not, it's the outer arrow that would be the pry point.

If it says "use wrench XYZ" that usually indicative of a one-piece case. You'd need to remove the crystal and drop the movement out from the front.