Proof Nehari manifold of semilineal subcritical $-\Delta u = f(u)$ in $\Omega$ is not empty.
If we only want to show that the Nehari manifold is not empty, we can do the following. Let $J_u(t)=J(tu)$ where $u\in H_0^1(\Omega)\cap L^\infty(\Omega)$ and $u>0$. As you alread have noted, we have that $$J'_u(t)=t^2\|u\|^2-\int f(tu)tu.$$
From 3. you have that $(1-\alpha)f(s)s\ge 2 F(s)$, which implies that $$\frac{F'(s)}{F(s)}\ge \frac{2}{(1-\alpha)s}.$$
For $s\geq1$, we conclude that $$F(s)\geq C_1s^{\frac{2}{1-\alpha}},$$
for some positive constant $C_1$. Therefore $F(s)\ge C_1s^{\mu}+C_2$ for $s>0$, $C_2\in\mathbb{R}$ a constant and $\mu>2$. By using again the inequality $(1-\alpha)f(s)\ge 2F(s)$, we have that $$-f(s)s\le -D_1s^\mu-D_2,\ s>0,$$
where $D_1>0$ and $D_2\in\mathbb{R}$ are contants. So $$J'_u(t)\le t^2\|u\|^2-D_1t^\mu\int u^\mu-D_2|\Omega|.$$
Once $\mu>2$, we must conclude that $J_u'(t)\to -\infty$ when $t\to \infty$.
On the other hand, the hypotheis $f'(0)$ implies that for small $x>0$, there is $\delta>0$ such that $$f(x)<(\lambda_1-\delta)x,$$
which implies that for small $x>0$ $$-f(x)x> -(\lambda_1-\delta)x^2.$$
Therefore $$J'_u(t)\ge t^2\|u\|^2-(\lambda_1-\delta)t^2\int u^2.$$
Once $\|u\|_2^2\le \frac{1}{\lambda_1}\|u\|$, we obtain that $$J'_u(t)\ge t^2\|u\|^2-\frac{\lambda_1-\delta}{\lambda_1}t^2\|u\|^2,$$
which implies that $J'_u(t)$ is positive near ht origin. Be cause of the continuity of $J'_u$, we conclude that there is $t>0$ such that $J'_u(t)=0$, which is to say that $$\langle J'(tu),tu\rangle =0,$$
or equivalently that $tu$ belongs to the Nehari manifold.
Edit: In the above calculations, I have assumed implicitly that $F(s)\ge 0$, which is not true, however, we can overcome this problem by using the hypothesis $f(s)/s\to \infty$ when $t\to \infty$, indeed, we can conclude that $$F(s)\ge C s^{\mu},$$
where $\mu$ is as above and $s$ is such that $F(s)=\int_0^s f(t)dt$ is positive (which will be true for bigger $s$). Therefore $F(s)\ge C s^{\mu}+D$ for all $s>0$, where $C>0$ and $D\in\mathbb{R}$ are constants.