The two "corollaries" are definitely false. The easiest counterexample I can think of is the following:
Consider the number 0.1234567891011121314151617181920212223...
which is constructed by basically just counting in base 10 past the decimal point. First, note that this number is irrational: every rational number has a repeating decimal representation after a certain point, yet that clearly never happens with this number; to be less vague, note that the string 1000...0001 for any finite number of zeros between the two 1s must appear eventually, and this wouldn't be the case for any rational number, since any repeating sequence of digits either trails off with infinitely many zeros (hence the 1 at the end wouldn't appear, unless something like 100...001 appears before the repeating part, but we can deal with that by just taking a sufficiently large number of zeros) or could at most contain only a certain finite number of zeros between two 1s (and for this irrational number, we can find arbitrarily many 0s between two 1s).
Now that we know that this number is irrational, we immediately see that the first corollary is false, since 10^(10^6) is a positive integer and its base 10 string will hence pop up eventually. We can see that the second corollary is false by looking at the same number, and noting that for any digit x in {0,1,...,9}, the string xxx...x (the digit x appearing a million times) must occur somewhere along the way by the same rationale.
I have no idea about pi and will think about that more later.