Those results are standard and appear in (almost) any functional analysis textbooks. The equivalence between (1), (2) and (3) appears in the text "Analysis Now" by Petersen (full proof with all the details).
$\bf Edit:$ $(1)\rightarrow (2)$ Suppose $T$ is continuous at $v_0$ and let $x\in X$ be given. Let $\epsilon>0$ be given. Find $\delta>0$ so that $\| T(v_0)-T(v)\|<\epsilon$ whenever $\| v_0-v\|<\delta$. Note that if $\|x-y\|<\delta$ then $\|x-y\|=\| v_0-(x-y+v_0)\|<\delta$. It follows from linearity that $T(x)-T(y)=T(v_0)-T(x-y+v_0)$, hence $\|T(x)-T(y)\|<\epsilon$. Thus $T$ is continuous at $x$.
$(2)\rightarrow (3)$: If $T$ is continuous at zero then there exists $\delta>0$ so that $\|T(x)\|\leq 1$ for all $\|x\|\leq \delta$. This implies that, for all $x\ne 0$, $\|T(\frac{\delta\cdot x}{\| x\|})\|\leq 1$, hence $\| T(x)\|\leq \frac{\| x\|}{\delta}$. Therefore $T$ is bounded.
$(3)\rightarrow (4)$: Follows easily from Beni's observation and $(4)\rightarrow (1)$ is obvious.