I am really struggling understanding the following equalities, particularly the last one. I think the first one is using the delta function, however I do not understand why the negative sign disappears too?
\begin{equation} -\frac{1}{2} \frac{\partial \text g^{ij}}{\partial q^k}\text g_{ia}\text g_{jp}\dot q^a\dot q^p=\frac 12\frac{\partial \text g_{ap}}{\partial q^k}\dot q^a\dot q^p=\frac{\partial \text g _{ak}}{\partial q^q}\dot q^a\dot q^q+\text g_{ak}\ddot q^a \end{equation} This is part of the geodesic equation. Your help is much appreciated!
Note that the indices of the metric tensor are raised in the left-most expression, but lowered in the middle expression. Recall that $$ g^{ij}\,g_{js} = \delta^i_s $$
Take the partial derivative with respect to $q^k$ of both sides and you'll find where the negative sign went.
As for the right-most expression, it's simply the middle one where the time term is separated out from the space terms, and written out explicitly.
Note the indices! Always pay attention to the indices! :)
On a side note, bad notation in that expression. Generally, Greek indices run from 0 to 3 and Latin indices from 1 to 3. That distinction makes it easier to see when the time index is written out explicitly and when not. In the expression you quoted, every index is Latin.