If $1\leq p\leq\infty$, then yes, both are norms.
The ordinary $\ell^p$ norm can decrease with $p$. E.g., Consider $(1,1)$ in $\mathbb{R}^2$. (And as Theo shows in his answer, the $\ell^p$ norm is always nonincreasing.)
Both norms can be seen as $L^p$ norms on $\mathbb{R}^n$ considered as functions on an $n$-point measure space. For the generalized mean, each point has measure $\frac{1}{n}$, so that the space is normalized to have measure $1$, making it a "probability space". For the ordinary $\ell^p$ norm, each point has measure $1$. It is true in general for probability spaces that the $L^p$ norm increases with $p$. This almost never holds for spaces of measure greater than $1$, as can be seen by taking a characteristic function of a set of finite measure greater than $1$. On the other hand, having the $L^p$ norm decrease with $p$ is also atypical, and happens only when there are no sets of positive measure less than $1$ (again consider a characteristic function).
Since I now see that you want to consider infinite sequences, I'll say a few words on generalized means for that case. For each sequence of weights $w_1,w_2,\ldots$ consisting of positive numbers summing to $1$, you can define the generalized mean with exponent $p$ by $(w_1|x_1|^p+w_2|x_2|^p+\cdots)^{1/p}$. The set of all sequences $(x_1,x_2,\ldots)$ with finite mean forms the $L^p$ space of a countable measure space with total measure $1$, if you like thinking of it that way. It will contain the ordinary $\ell^p$ space.