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$\newcommand\si\sigma$Yes.

Indeed, $$F_p:=F_p(m,\si)=\|(\si-1)Z+mR\|_p,$$ where $Z$ and $R$ are independent random variables, $Z$ is standard normal, $R$ is Rademacher (with $P(R=\pm1)=1/2$), and $\|X\|_p:=(E|X|^p)^{1/p}$.

So, the desired result follows by the central limit theorem and the Haagerup inequalities, which imply that, for each real $p>0$, $$G_p:=\Big\|\sum_{i=1}^n a_iR_i\Big\|_p$$ differs from $G_2$ by at most a factor depending only on $p$, where the $a_i$'s are any nonzero real numbers and the $R_i$'s are independent Rademacher r.v.'s.

$\newcommand\si\sigma$Yes.

Indeed, $$F_p:=F_p(m,\si)=\|(\si-1)Z+mR\|_p,$$ where $Z$ and $R$ are independent random variables, $Z$ is standard normal, $R$ is Rademacher (with $P(R=\pm1)=1/2$), and $\|X\|_p:=(E|X|^p)^{1/p}$.

So, the desired result follows by the central limit theorem and the Haagerup inequalities, which imply that, for each real $p>0$, $$G_p:=\Big\|\sum_{i=1}^n a_iR_i\Big\|_p$$ differs from $G_2$ by at most a positive real factor depending only on $p$, where the $a_i$'s are any nonzero real numbers and the $R_i$'s are independent Rademacher r.v.'s.

$\newcommand\si\sigma$Yes.

Consider first the case $p\ge1$. Then $$F_p:=F_p(m,\si)=\|(\si-1)Z+mR||_p\le|\si-1|\,\|Z\|_p+|m| \le\|Z\|_p\sqrt2\, F_2,$$ where $Z$ and $R$ are independent random variables, $Z$ is standard normal, $R$ is Rademacher (with $P(R=\pm1)=1/2$), and $\|X\|_p:=(E|X|^p)^{1/p}$.

On the other hand, again for $p\ge1$, by Jensen's inequality, conditioning on $R$ and then on $Z$, we get $$F_p\ge\max(|\si-1|\,\|Z\|_p,|m|)\ge\frac{F_2}{\sqrt2}.$$

Now consider the case $p\in(0,1)$. Then, by Jensen's inequality, $$F_p\le F_2.$$ Finally, again for $p\in(0,1)$, by the central limit theorem and the Haagerup inequality, $$F_p\ge2^{1/2-1/p}F_2.$$

$\newcommand\si\sigma$Yes.

Indeed, $$F_p:=F_p(m,\si)=\|(\si-1)Z+mR\|_p,$$ where $Z$ and $R$ are independent random variables, $Z$ is standard normal, $R$ is Rademacher (with $P(R=\pm1)=1/2$), and $\|X\|_p:=(E|X|^p)^{1/p}$.

So, the desired result follows by the central limit theorem and the Haagerup inequalities, which imply that, for each real $p>0$, $$G_p:=\Big\|\sum_{i=1}^n a_iR_i\Big\|_p$$ differs from $G_2$ by at most a factor depending only on $p$, where the $a_i$'s are any nonzero real numbers and the $R_i$'s are independent Rademacher r.v.'s.

$\newcommand\si\sigma$Yes.

Consider first the case $p\ge1$. Then $$F_p:=F_p(m,\si)=\|(\si-1)Z+mR||_p\le|\si-1|\,\|Z\|_p+|m|,$$ where $Z$ and $R$ are independent random variables, $Z$ is standard normal, $R$ is Rademacher (with $P(R=\pm1)=1/2$), and $\|X\|_p:=(E|X|^p)^{1/p}$.

On the other hand, again for $p\ge1$, by Jensen's inequality, conditioning on $R$ and then on $Z$, we get $$F_p\ge\max(|\si-1|\,\|Z\|_p,|m|)\ge\frac12\,(|\si-1|\|Z\|_p+|m|).$$

Now consider the case $p\in(0,1)$.

Then, by Jensen's inequality, $$F_p\le F_2=(|\si-1|^2+|m|^2)^{1/2}.$$

Finally, for $p\in(0,1)$, by the central limit theorem and the Haagerup inequality, $$F_p\ge2^{1/2-1/p}F_2.$$

$\newcommand\si\sigma$Yes.

Consider first the case $p\ge1$. Then $$F_p:=F_p(m,\si)=\|(\si-1)Z+mR||_p\le|\si-1|\,\|Z\|_p+|m| \le\|Z\|_p\sqrt2\, F_2,$$ where $Z$ and $R$ are independent random variables, $Z$ is standard normal, $R$ is Rademacher (with $P(R=\pm1)=1/2$), and $\|X\|_p:=(E|X|^p)^{1/p}$.

On the other hand, again for $p\ge1$, by Jensen's inequality, conditioning on $R$ and then on $Z$, we get $$F_p\ge\max(|\si-1|\,\|Z\|_p,|m|)\ge\frac{F_2}{\sqrt2}.$$

Now consider the case $p\in(0,1)$. Then, by Jensen's inequality, $$F_p\le F_2.$$ Finally, again for $p\in(0,1)$, by the central limit theorem and the Haagerup inequality, $$F_p\ge2^{1/2-1/p}F_2.$$