The first one is Dvoretzky's theorem (or, more precisely, a consequence of Dvoretzky's theorem), which says not only that you can find $1+\epsilon$ isomorphic copies of $\ell_2^n$ in infinite dimensional spaces, but that you can find a $1+\epsilon$ copy of $\ell_2^n$ in any spaces whose dimension is at least $N$, where $N$ is a function of $\epsilon$ and $n$. The proof is technical, but the idea is simple. We use the idea of concentration of measure. We consider the norm $\|\cdot\|$ on the $N$-dimensional space $\mathbb{R}^N$. By John's theorem, after applying a suitable invertible linear translation, we can assume the Lipschitz constant of $\|\cdot\|$ with respect to the $\ell_2^n$ norm of $\mathbb{R}^N$ is at most $N^{1/2}$. Then one can show that, since the function $\|\cdot\|$ is concentrated around its mean (or median, depending on which version of concentration you use), there is a very small measure exceptional set of the sphere such that on the complement of the exceptional set, the norm $\|\cdot\|$ is almost exactly a fixed multiple of the $\ell_2^N$ norm. This exceptional set is small enough to be able to find an $n$-dimensional subspace which contains none of the exceptional set, and therefore the $\ell_2$ and $\|\cdot\|$ norms are almost exactly a multiple of each other on this $n$-dimensional subset.

Alternatively, if you know Krivine's theorem (which I personally find easier to understand), then you can deduce Dvoretzky's theorem. Every infinite dimensional Banach space has a basic sequence. There exists $1\leqslant p\leqslant \infty$ such that we can find $1+\epsilon$ $\ell_p^n$ spaces in the sequence (and even as blocks of the sequence). If $p=\infty$, this means $c_0$ (and by your second question, every space) is finitely representable in your space, including $\ell_2$. If $1\leqslant p<\infty$, this means $\ell_p$ is finitely representable in your space. But since $L_p$ is finitely representable in $\ell_p$, this means $L_p$ is finitely representable in your space. And $L_p$ contains an isometric copy of $\ell_2$ as the closed span of independent gaussian random variables. This is a roundabout way, but it contains a lot of interesting pieces along the way.