Question (1). Riesz-Fischer Theorem: For $1\leq p\leq \infty$, $L^p(\mu)$ is complete.
Corollary of proof: Let $1\leq p\leq \infty$. If $(f_n)_{n=1}^\infty$ is a sequence coverging to $f$ with respect to $||\cdot||_p$ in $L^p(\mu)$, then there exists a subsequence $(f_{n_k})_{k=1}^\infty$ such that $f_{n_k}$ converges to $f$ a.e. in $\Omega$.
Do these results hold for $0<p<1$?
Question (2). Let $p>1$. Let $h$ be nonnegative, $h\in L^p(\mathbb{R}^n)$. Let $h^*(x)=\sup_{x\in B}\dfrac{\int_B|h|d\mu}{\mu(B)}$ be the maximal function of $h$. Then $||h^*||_p\leq C||h||_p$, where $C=(\dfrac{3^n2p}{p-1})^{1/p}$.
Is there any estimate of the form $||h^*||_p\leq C||h||_p$ for $0<p\leq 1$?
I can only say something about (1) at the moment. In particular, for $0 < p < 1$, $L^p$ is not even a metric space because the triangle inequality fails so it cannot be a complete normed space.