The first fact follows trivially from the fact that the digit sum equals the total number of digits, which is equal to the base, from the definition of self-descriptive number.
That a self-descriptive number in base b must be a multiple of that base (or equivalently, that the last digit of the self-descriptive number must be 0) can be proven by contradiction as follows: assume that there is in fact a self-descriptive number m in base b that is b-digits long but not a multiple of b.
If x > 1, then m has more than b digits, leading to a contradiction of our initial statement.
Note that this depends on being allowed to include as many trailing zeros as suit, without them adding any further information about the other present digits.
Considering a hypothetical case where the digits are treated in the opposite order: the units is the count of zeros, the 10s the count of ones, and so on, there are no such self-describing numbers.
Attempts to construct one result in an explosive requirement to add more and more digits.