Summary: The inequality $F(x)\geqslant0$ does not hold for every admissible values of the parameters $x$, $a$ and $b$, but replacing the last $12$ by $6$, or $a^2x^2$ by $ax$, both make it true.
1. The change of variables $ax/b\to s$, $1/b\to z$ shows that the inequality to prove is equivalent to
$$
U_s(z)\geqslant z^2-s^2\tag{$\ast$}
$$
for every $0\lt s\lt z$, where
$$
U_s(z)=u(s+z)-u(s)+u(2s-z)-u(2s),
$$
and
$$
u(t)=t\phi(t)+(1+t^2)\Phi(t).
$$
2. Since $\Phi'(t)=\phi(t)$ and $\phi'(t)=-t\phi(t)$, one has $u''(t)=2\Phi(t)$ and
$$
U_s'(z)=u'(s+z)-u'(2s-z)=\int_{2s-z}^{s+z}u''(t)\mathrm dt=\int_{2s-z}^{s+z}2\Phi(t)\mathrm dt.
$$
3. Consider the asymptotics $s=o(1)$ and $z=rs$, with $r\gt1$ fixed. Then $2\Phi(0)=1$ and $(s+z)-(2s-z)=(2r-1)s$ hence $U_s'(rs)=(2r-1)s+o(s)$ when $s\to0$, for every fixed $r\gt1$. Since $U_s(s)=0$, one gets
$$
U_s(rs)=\int_1^rsU_s'(ws)\mathrm dw=s^2\int_1^r(2w-1)\mathrm dw+o(s^2)=(r^2-r)s^2+o(s^2).
$$
Since $z^2-s^2=(r^2-1)s^2$ and $r^2-1\gt r^2-r$, one sees that $(\ast)$ is wrong in the regime $z=rs$, $r\gt1$ fixed, for $s$ small enough.
4. On the other hand, at least in the regime $z=rs$, $r\gt1$ fixed, $U_s(z)\geqslant z(z-s)+o(s^2)$ hence $U_s(z)\geqslant\frac12(z^2-s^2)+o(s^2)$. Reformulating everything as in the question, this shows that one can still hope that,
for every $b\gt0$, $0\lt a\lt 1$ and $0\lt x\lt1/a$,
$\qquad[12b^2(g(\frac{1+ax}{b})-g(\frac{ax}{b})+g(\frac{2ax-1}{b})-g(\frac{2ax}{b}))] +$
$\qquad\qquad\qquad +{}[12b^2(\Phi(\frac{1+ax}{b})-\Phi(\frac{ax}{b})+\Phi(\frac{2ax-1}{b})-\Phi(\frac{2ax}{b})] \geqslant\mathbf{6}(1-a^2x^2),
$
5. Indeed, $U_s(z)\geqslant z(z-s)$ holds true for every $0\lt s\lt z$. To show this, start from the identity $\Phi(-t)=1-\Phi(t)$ and the change of variable $\tau=-t$ in the formula of $U_s'(z)$ above, which yield
$$
U_s'(z)=\int_{-s-z}^{-2s+z}2(1-\Phi(\tau))\mathrm d\tau=2(2z-s)-\int_{-s-z}^{-2s+z}2\Phi(t)\mathrm dt,
$$
hence
$$
U_s'(z)=2(2z-s)-\int_{2s-z}^{s+z}2\Phi(t-3s)\mathrm dt.
$$
Since $\Phi(t-3s)\leqslant\Phi(t)$ for every $t$, this yields
$$
U_s'(z)\geqslant2(2z-s)-\int_{2s-z}^{s+z}2\Phi(t)\mathrm dt=2(2z-s)-U_s'(z),
$$
hence $U_s'(z)\geqslant2z-s$ and, finally,
$$
U_s(z)\geqslant z^2-zs.
$$
In the notations of the question, this proves that
$[12b^2(g(\frac{1+ax}{b})-g(\frac{ax}{b})+g(\frac{2ax-1}{b})-g(\frac{2ax}{b}))] +$
$\qquad\qquad\qquad +{}[12b^2(\Phi(\frac{1+ax}{b})-\Phi(\frac{ax}{b})+\Phi(\frac{2ax-1}{b})-\Phi(\frac{2ax}{b})] \geqslant\mathbf{12}(1-ax)\geqslant\mathbf{6}(1-a^2x^2).
$