How important is it to re-space my rear hub? I just bought new wheels for an older Schwinn frame. I don't know the date it was made, but it's a steel frame and as built it had stem shifters and a cup-and-cone bottom bracket, etc. Probably from some time in the 80s, I estimate.

The spacing between the rear dropouts on the frame itself is 130mm; the new rear wheel and hub arrived spaced to 120mm. It's a flip flop hub with a solid axle (no quick release). I am able to get the hub to fit snugly between the dropouts just by tightening the axle nuts, since the steel frame will bend to make up that 5mm on each side pretty easily. 

Is this a safe setup? Or do I need to add spacers to the hub to fit the 130mm frame without any bending?
No, that much of a spacing difference is generally a bad idea even with steel because of the increased stress it causes in the dropout area, which may cause or contribute to failure of the frame in the dropout area over time.

How bad it is exactly on a given frame is difficult to pin down. But it's observably true that the right rear dropout is a common failure point from fatigue on heavily ridden bikes where everything else is holding up well. The static stress that gets created from having the frame squeezed down can only add to that.

Presuming it's steel, just respace it to 120, and then re-align the dropouts afterward. A side benefit is that you get to align the rear end at the same time you respace it, so it can be perfect on both fronts.

Having the spacing and dropout alignment be right will also make tensioning the chain during wheel re-installation be a much smoother process.