Can magnets damage a CRT: yes.
Do most fix themselves: yes.
Most modern CRT's automatically 'degauss' at turnon. Your old screen may not have that feature.
A large looped wire with at least six loops with an AC current flowing in it will fix your screen.
You can use a LONG extension cord and make it into a perfect circle 20" diameter. At least six loops, better ten, and let it be in use running some big current draw item.
the loops should be rigid!! use a coathanger wire as a form.
Take this contraption and bring it (while powered) near you TV screen.. moved it slowly!! slowly around and around the turned on TV screen. VERY close! 1 inch away. Then slowly very slowly move the coil away from the screen while both have current running through them. DO NOT TURN OFF THE COIL NEAR THE SCREEN!!!!!!!!!!!!!!!!
This should fix the worst old TV magnetization.