If infected or frostbite on fingers then you would have to get hand cut off! Is that true?

Not usually. Typically the smaller blood vessels are affected and so most cases of amputation for frostbite involve only the tips of the fingers or first knuckle. Severe frost bite could theoretically cause someone to lose their hand but usually if hypothermia is that bad you would not survive.
Not likely. While it is possible that a finger infection might spread enough that a hand amputation would be necessary, it is very unlikely.