How likely is it that I was told my 1989 Honda Civic (336K miles) has two broken motor mounts, broken axle (not CV boot, but axle)? There is minor vibration from front end, no clunking or loud noises & drives fine.
I think what you are actually asking; Is it likely that your 336K Civic NEEDS 2 motor mounts and an axle?
I’d say, very likely.
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They could have found the motor mounts during the inspection and they could be the cause of a vibration, especially at idle. But I can’t imagine you would have an actual broken axle shaft or tube without serious problems driving. I would ask to see that with your own eyes before you pay for that repair.
I agree with the number-name guy. 
That car is not equipped with an LSD unless it’s a Japanese-market version that was optioned with it. Without an LSD, if one axle breaks, the other won’t turn. With an LSD, if one axle breaks (like, actually breaks into 2 pieces) the other will turn briefly before you fry the LSD. Either way, by now the car wouldn’t be able to move if the axle was broken. Are you sure this is what they said was broken?
How much are repairs vs value of car? And doing no repairs and buying another car? $600 is 2 car payments. But buying a newer car is something you may have to do eventually.