How lovely!
I agree with the above comments. To answer a couple of your queries, the movement is a 33.3 T3 (which first appeared in 1940). It was the first movement that had two ‘dedicated’ buttons (the earlier T1 had the reset button through the crown, the T3/yours had a separate reset button at 4 o’clock. The calibre is marked under the balance wheel (we can see the ’33.3’ - you should be able to see ‘T3’ under the balance).
The ‘UNADJUSTED’ mark shows that it was intended for the US market.
If you can get hold of a copy of the Omega book ‘A Journey Through Time’ you will find quite a bit about the 33.3 (but not specifically your watch).
Yes, I would recommend that you get an Extract. It will be of interest to you - but also of value if you sometime wanted to sell the watch.