Replacing Capacitor with a slightly less capacitance

The value of the capacitor is not that critical, it does not effect the resonant frequency of the magnetron. Using that capacitor will give you slightly less power but you wont notice the difference. The voltage rating should be at least as much as the origional capacitor. I repaired Microwave ovens for 30 years.

That 1.2 uf cap is in a tuned circuit (with inductor) which has to do with frequency and should be replaced with same value. The replacement will probably work since values of components will change slightly over time but I would still stick to the original value.

No, you need to use one that has the same specs as the old one. You should be able to find one online with the model number that's on the old one, or perhaps contacting the manufacturer for the part number. Circuits are designed with certain specs, that if altered, could cause poor performance (not as much power as before) or burn something else out. Digi-key is a good place to look for electronic parts too.

Personally, being that it is a microwave I would have no problem changing it out with that cap. 0.06uF should be within tolerance. Worst case you may be putting additional wear on something else down the line.