1 is very hard and 2 is very easy. For 2, take an $\varepsilon$-net $(x_i)_{i=1}^m$ in the sphere of the $n$ dimensional space ($m$ depends on $n$ and $\varepsilon$), and norming functionals $f_i$'s. Now check that the map from the $n$-dimensional space into $\ell_{\infty}^m\subset c_0$ given by $x\to (f_1(x), f_2(x), \ldots, f_m(x))$ works.

For 1 there is a bit of easier proof if you don't care about the constant being $1+\varepsilon$ and if you are familiar with spreading models but still pretty involved. But standard reference for 1 is Milman-Schechtman book

1 is very hard and 2 is very easy. For 2, take an $\varepsilon$-net $(x_i)_{i=1}^m$ in the sphere of the $n$ dimensional space ($m$ depends on $n$ and $\varepsilon$), and norming functionals $f_i$'s. Now check that the map from the $n$-dimensional space into $\ell_{\infty}^m\subset c_0$ given by $x\to (f_1(x), f_2(x), \ldots, f_m(x))$ works.

For 1 there is a bit of easier proof if you don't care about the constant being $1+\varepsilon$ and if you are familiar with spreading models but still pretty involved. But standard reference for 1 is Milman-Schechtman's book