Proof by induction of Bernoulli's inequality: $(1 + x)^n \geq 1 + nx$

$$\begin{align} (1 + x)^{k+1} & \geq (1+kx)(1+x) \\ \\ & = 1 + kx + x + kx^2 \\ \\ & = 1 + (k+1)x + kx^2 \\ \\ & \geq 1+ (k+1) x\end{align}$$ as desired.

Remark: The inductive hypothesis $P(k)$ is assumed only for some arbitrary $k \in \mathbb N$; i.e., the point of an inductive proof is to then show that it is indeed true for all $n \in \mathbb N$, by first showing that $P(1) \land (P(k) \implies P(k+1)).$

You're almost done. $(1+x)^{k+1}\geq (1+kx)(1+x)=1+kx^2+(k+1)x\geq1+(k+1)x$ since $kx^2\geq 0$.

Also in your assumption, "$1+x>0\implies (1+x)^k\geq 1+kx,~\forall~k\in\mathbb{N}\setminus\{1\}$", you shouldn't write $\forall~k\in\mathbb{N}\setminus\{1\}$. Because if you assume this is true for all $k\in \mathbb{N}$, then you've already assumed it is true for $k+1$. Also, you shouldn't let $k\ne 1$, since then your argument doesn't allow you to conclude $P(1) \implies P(2)$.

You have $(1 + x)^{k+1} \geq 1 + (k+1)x + kx^2$. Keeping in mind that $kx^2 \geq 0$, what does this imply about your inequality?