Donor cables hooked up correctly, while the receiving car cables were reversed. It didn't start, but smoked instead. Mechanic shop used a portable jumper, then used receiving car again and had cables reversed. Again it smoked and didn't start. Then a friend checked and said alternator fuse had blown. They replaced it with a 75A instead off 100A fuse with instructions to replace right away back to the normal amperage fuse. They checked the battery and alternator with machine and they checked out ok. I took the car to a shop and replaced alternator fuse and radio fuse. They said the battery was shorted and alternator was no good. Is this possible with regard to battery and alternator?
It's entirely possible to burn out the alternator and battery by hooking the jumper box (or jumper cables) up backwards. The alternator can fry rather quickly (diodes will fry and the alternator becomes useless). The battery can take longer.
Basically when you hook a battery up in series (backwards) instead of parallel (normal), you create a circuit. This will not only create sparks, it generates a lot of heat which you'll figure out very quickly through the jumper cables. Putting them on backwards can fry the battery if left there long enough.
I've accidently had the the battery cables hooked up wrong once, but I quickly removed them and redid it with no ill affects.
If you don't check them regularly, like once a year, the battery/alternator hookups could affect each other with one causing the other to go out sooner. I've had it happen several times. You can buy a simple battery/alternator check device, I think for less than $10 USD.
This is what I use to check both twice a year, but if you took it somewhere and it checked out on a machine, I would go with that over the verbal of a shop, unless you really know their reputation.