Calculating the round metric on $S^n$

Question

I'm self-studying differential geometetry from Taubes' textbook on Differential Geometry and I found a lot of unclear things in the following passage:

Taubes gives a formula for $\Gamma^i_{jk}$ but doesn't give the derivation yet and expects us to use it. $g^{ij}$ denotes the i,jth entry in the inverse matrix of $g$. [

I have a few questions:

1. How does he calculate the pullback of $g_{ij}$? Because I got a different answer $g_{ij} = \delta_{ij} + y_i y_j (1-||y||^2)^{-1}$
2. How does he calculate the $\Gamma^i_{jk}$? I can't even begin to calculate it, as it requires me to find the inverse matrix of $g$, a task which seems far too complicated in this case.
3. How come the equation with the $\mathcal{O}(|y|^2)$ is actually equivalent to the formula he gave before? His argument is incomprehensible to me.
4. How does the formula imply that $t \to x_j(t)$ lies on a plane?

These questions are enough for now.