Why do we miss 8 in the decimal expansion of 1/81, and 98 in the decimal expansion of 1/9801? I've

For $\frac{1}{81}$, there was an 8, but it got bumped up. We can write $\frac{1}{81}$ as this sum:
$\frac{1}{81}=\begin{array}{rl}& 0.0\\ +& 0.01\\ +& 0.002\\ +& 0.0003\\ +& 0.00004\\ +& 0.000005\\ +& 0.0000006\\ +& 0.00000007\\ +& 0.000000008\\ +& 0.0000000009\\ +& 0.00000000010\\ +& 0.000000000011\\ +& \underset{\_<!--\; \_\; -->}{⋮<!--\; ⋮\; -->}\\ & 0.01234567901\dots <!--\; \dots \; -->\end{array}$
This kind of effect of "carrying the 1" when the nine digit flips to a ten is the thing that is causing the behavior in all of the fractions you are describing.
To follow-up, this should provide a bit more insight as to why interesting patterns appear in the decimal representations of fractions with a power of 9 or 11 as the denominator, and see why we can write those numbers like $\frac{1}{81}$ as that sum. First note that
$\frac{1}{9}=(\frac{1}{10}+\frac{1}{100}+\frac{1}{1000}+\cdots <!--\; \cdots \; -->)$
so if we were to consider $\frac{1}{81}$ like before, we would have
$\frac{1}{81}={\left(\frac{1}{9}\right)}^2={(\frac{1}{10}+\frac{1}{100}+\frac{1}{1000}+\cdots <!--\; \cdots \; -->)}^2$
Then if we were to want to know the value of the ten-thousandth's decimal place of $\frac{1}{81}$ we would just have to find the numerator of the term in the expansion of this square with a denominator of 10000, which we can readily see is
$\begin{array}{rl}\frac{1}{81}=(\cdots <!--\; \cdots \; -->+((\frac{1}{10})(\frac{1}{1000})+(\frac{1}{100})& (\frac{1}{100})+(\frac{1}{1000})(\frac{1}{10}))+\cdots <!--\; \cdots \; -->)\\ =(\cdots <!--\; \cdots \; -->+(\frac{3}{10000}& )+\cdots <!--\; \cdots \; -->)\end{array}$
So it looks like these evident patterns that appear in the decimal expansions of fraction with a multiple of 9 in the denominator is due at least partly to the fact that this infinite sum representation of $\frac{1}{9}$ consists entirely of terms with a numerator of 1, so multiplying this sum into things may result in "predictable" behavior that results in a pattern.
As for why having a multiple of 11 in the denominator makes similar patterns, note that
$\begin{array}{rl}\frac{1}{11}& =.090909090909090909\dots <!--\; \dots \; -->\\ & =(\frac{9}{100}+\frac{9}{10000}+\cdots <!--\; \cdots \; -->)\\ & =(\frac{10-<!--\; -\; -->1}{100}+\frac{10-<!--\; -\; -->1}{10000}+\cdots <!--\; \cdots \; -->)\\ & =(\frac{1}{10}-<!--\; -\; -->\frac{1}{100}+\frac{1}{1000}-<!--\; -\; -->\frac{1}{10000}+\cdots <!--\; \cdots \; -->)\end{array}$
So again we have an infinite sum of terms each with a numerator of 1 (just alternating sign this time) that will result in certain "predictable" patterns when multiplied by things.