Decimal Expansion of Pi

The claim is only about finite strings (and apart from this, it is only conjectured, has not been proven). In fact your what-if argument is sound and would show that $\pi$ is rational. The fact that it is not rational (in fact, transcendental) shows that it cannot contain itself in a nontrivial manner.

Regarding the second question: No, all finite strings does not imply a given infinite string. In fact, the number $$0.123456789101112131415161718192021222324\ldots $$ obtained by concatenating all natural numbers provably contains every finite string, and among these $3$, $31$, $314$, $3141$ and so on, but certainly (though perhaps not obviously) not the full expansion of $\pi$.

There are a number of observations that would lead to the conclusion that having every finite string as a substring is totally different from having every infinite string as a substring.

Firstly, "$100111000011111000000...$" contains (as a substring) every finite string consisting of only ones or only zeroes, but it does not contain the infinite strings consisting of only ones or only zeroes.

Secondly, concatenating all positive integers yields "$12345678910111213...$" that contains every positive integer but does not contain the infinite string "$0000...$" because every positive integer has finitely many zeroes. This is a much easier statement to verify than Hagen's claim that it does not contain $π$.

Thirdly, the number of substrings that a string contains is countable, and the number of infinite strings is uncountable, so any given string will not contain almost all infinite strings.

Fourthly, your attempt to justify your hypothesis is logically flawed in a crucial way. If an infinite string $x$ contains every finite string, it means:

  For every finite string $y$:

    For some position $p$:

      $y$ occurs in $x$ at position $p$.

It does not imply:

  For some position $p$:

    For every finite string $y$:

      $y$ occurs in $x$ at position $p$.

which is what you would need to conclude that:

  For some position $p$:

    $π$ occurs in $x$ at position $p$.

This switching of quantifiers is an extremely common logical error but it should be very obvious if you wrote it out the way I did.