Almost sure convergence of a sequence of random variables and convergence in the $r$th mean (for any $r$) each imply convergence in probability. However, (I believe) neither of them implies the other for fixed $r$.
What about convergence in the $r$th mean for all $r$? This implies almost sure convergence, correct?
On
This is not true, because of the following lemma (cf. Wikipedia):
If $(X_n)_n$ converges in probability to $X$, and if $\mathbb{P}\{|X_n| \leq b\} = 1$ for all $n$ and some $b$, then $(X_n)_n$ converges in $r$th mean to $X$ for all $r \geq 1$.
In particular, this applies to the bounded sequence $(X_n)_n$ with $X_n$ being a Bernoulli with parameter $1/n$. But converges in probability to $0$, yet not almost surely.
The answer is negative.
Consider $\Omega = [0,1]$ with Lebesgue measure. Put $A_1 = [0,1]$, $A_2 = [0, \frac12]$, $A_3 = [\frac12, 1]$, $A_4 = [0, \frac14]$. $A_5 = [\frac14, \frac12]$, $A_6 = [\frac12, \frac34]$, $A_7 = [\frac34,1]$, $A_8 =[0,\frac18]$ and so on. Put $f_n = I_{A_n}$. Hence $f_n \to 0$ in every $L_r$, $r \ge 1$ and $f_n$ doesn't converge to $0$ (or anywhere else) a.s.