Can simple skin fungal infection become deadly in immune-compromised people?

Potentially. If the 'fungus' is untreated; and the person is indeed immunocompromised then potentially yes; rare, but it occurs.
Yes, but... ...The immune compromise would have to be very severe. Any fungal, bacterial, or viral infection that is normally self-limited and benign in the presence of a normal immune system can become a very serious infection if the immune system is damaged enough.