For a number to be irrational, it must be impossible to express the value as a ratio of integers. So, if I look at the infinite string of digits to the right of the number's decimal point, can I find any given integer sequence of length n (n natural), or is that not guaranteed?
It is not guaranteed. For example, $0.101001000100001...$ is irrational. A number is rational if and only if its decimal expansion either terminates or from some point on repeats, so any non-terminating decimal that doesn’t end in an infinite string of repetitions of a single finite block of digits must represent an irrational number.