= Solution
Write $F=f_2$. Split the integral in the assumed tail bound according to whether $g\leq a/2$ or $g>a/2$:
$$
a\mu\{F>a\}\leq\frac a2\mu\{F>a\}+\int_{\{g>a/2\}}g\,d\mu.
$$
The finite-superlevel hypothesis permits subtraction of the first term. With $a=2b$, this becomes
$$
b\mu\{F>2b\}\leq\int_{\{g>b\}}g\,d\mu.
$$
Thus $F/2$ satisfies part (i), which proves finiteness and the preliminary estimate
$$
\boxed{\|F\|_p^p\leq2^p q\|g\|_p^p}.
$$
Now integrate the original tail bound against $pa^{p-2}$. The <layer cake representation> and <Tonelli theorem> give
$$
\|F\|_p^p\leq q\int_XgF^{p-1}\,d\mu\leq q\|g\|_p\|F\|_p^{p-1},
$$
where the second step is <Hölder's inequality>. The preliminary estimate justified the finiteness needed here. If $\|F\|_p=0$ the result is immediate; otherwise divide by its $(p-1)$st power. This proves the <Lp bound from a tail domination inequality>:
$$
\boxed{\|F\|_p\leq q\|g\|_p,\qquad \|F\|_p^p\leq q^p\|g\|_p^p}.
$$
The printed strict signs cannot hold when $g=F=0$: both sides are zero. The universally valid formulation therefore has non-strict inequalities. If $\|g\|_p>0$, the sharper estimate is actually strict, as follows.
Suppose equality held and $\|F\|_p>0$. Equality in <Hölder's inequality> and the norm relation force $g=F/q$ <almost everywhere>; equality in the preceding integrated tail inequality forces equality in that tail inequality for almost every $a>0$. Put $H(a)=\mu\{F>a\}$ and $J(a)=\int_a^\infty H(t)\,dt$. The layer-cake identity on $\{F>a\}$ gives $\int_{\{F>a\}}F=aH(a)+J(a)$, so equality implies
$$
J(a)=(q-1)aH(a)\quad\text{for almost every }a>0.
$$
Because $F\in L^p$ and $p>1$, $J$ is finite for positive $a$, locally absolutely continuous, and $J'=-H$ <almost everywhere>. It follows that $J'=-(p-1)J/a$, whose solutions on $(0,\infty)$ are $J(a)=c a^{1-p}$. If $c>0$, then $H(a)=c(p-1)a^{-p}$ almost everywhere, and $p\int_0^\infty a^{p-1}H(a)\,da$ diverges. If $c=0$, then $F=0$ almost everywhere. Both contradict the assumed nonzero finite equality case. If $F=0$ but $g\ne0$, strictness is immediate.
Thus both requested strict estimates hold when $\|g\|_p>0$: the coarse constant is strictly larger than the sharp one since $q^{1/q}\leq e^{1/e}<2$, so $q^p<2^p q$. The zero-data exception is the only qualification needed here.
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