How are molybdenum foil seals made? What’s the thickness of the foil? Also, how do they get the knife edge on the foil before making the seal?

The molybdenum foil is made from wire.  First the wire is flattened, and then the edges are rolled to reduce their thickness.  With the better quality foils, they are then electrochemically etched to further sharpen the knife-edges.  The thickness is about 0.01 to 0.05mm.  The seals are made by first welding a metal wire or electrode to the ends of the foil, and then embedding it in quartz.  The quartz is heated in excess of 2000C with an inert atmosphere inside the tube (vacuum, or an argon flush).  When it is soft it is either collapsed around the moly foil due to the vacuum inside the tube, or in case of atmospheric pressure seals made under argon gas the softened quartz is pressed around the foil using a set of mechanical pinch hammers.

Are the foils possible to handle/store or they must be made immediately before use ?

I suppose the pin-to-foil assemblies could be stored for a long time if kept sufficiently clean.  For some special lamps made in low volumes which required foils of special dimensions we still used spools of moly ribbon more than 20 years old without any notable problems.  However after welding pin-to-foil assemblies those were always stored in nitrogen-filled cupboards.

Perhaps the concern is that once they are welded together, any moisture from the air that condenses and touches the weld would cause corrosion from dissimilar metals ?

Perhaps the concern is that once they are welded together, any moisture from the air that condenses and touches the weld would cause corrosion from dissimilar metals ?

This is only a half-truth, because dissimilar metal corrosion is generally a slow process. Al on steel in salty air takes a while to get to work, and I would expect a lamp factory to have better working conditions.