1 to the power of infinity, why is it indeterminate?
It isn’t: $\lim_{n\to\infty}1^n=1$, exactly as you suggest. However, if $f$ and $g$ are functions such that $\lim_{n\to\infty}f(n)=1$ and $\lim_{n\to\infty}g(n)=\infty$, it is not necessarily true that
$$\lim_{n\to\infty}f(n)^{g(n)}=1\;.\tag{1}$$
For example, $$\lim_{n\to\infty}\left(1+\frac1n\right)^n=e\approx2.718281828459045\;.$$
More generally,
$$\lim_{n\to\infty}\left(1+\frac1n\right)^{an}=e^a\;,$$
and as $a$ ranges over all real numbers, $e^a$ ranges over all positive real numbers. Finally,
$$\lim_{n\to\infty}\left(1+\frac1n\right)^{n^2}=\infty\;,$$
and
$$\lim_{n\to\infty}\left(1+\frac1n\right)^{\sqrt n}=0\;,$$
so a limit of the form $(1)$ always has to be evaluated on its own merits; the limits of $f$ and $g$ don’t by themselves determine its value.
The limit of $1^{\infty}$ exist:$$\lim_{n\to\infty}1^n$$ is not indeterminate. However$$\lim_{a\to 1^+,n\to\infty}a^n$$ is indeterminate..
There are many reasons. For example, let $1^\infty=1$. Taking logarithm, you have $\infty\cdot 0=0$. Similarly for other operations you will obtain some absurd.