Δαφνη Ευτυχια Αρθουρος

the locked-up situation usually means that one or more sets of poles has a crud or connection issue - that is, when you switch on the motor and that particular set of poles is supposed to be in contact to set up the "go!" field, a connection somewhere isn't being made, and so no field is erected. I see that your rotor's pole contacts look all shiny, but I can't tell whether you cleaned off the stator's pole contacts - if those have crud on them, you may still stall on startup due to a lack of field.

Oil: axle grease, or any of that nature of lubes. Basically, a petroleum oil product that is more solid than liquid. the 10W30 should be fine for seeing whether the motor will work long-term, but if you're going to use it long-term you'll want to pack both axle bearings with the real deal.

As for the squeaking, and whether this fan has bit the dust: If after all the electrical connections are lovely and clean, the thing still won't consistently run smoothly or for more than a few minutes then the years of work and the cleaning may have taken off enough of the electrical contact area between the stator and the rotor that the motor just isn't going to do much any more. Can't say for sure but that's what it sounds like to me.

Light machine oil, sewing machine oil will do. 3-in-one oil is fine. But you don't
want a water displacer like WD (guess what WD stands for) or food grade oil (although
my chainsaw uses vegetable oil for bar lube).