Mathematically speaking I do not understand why from
\begin{equation} T(q^{\lambda},\dot{q}^{\lambda},t)= \dfrac{1}{2} \,g_{\mu \nu} \, \dot{q}^{\mu} \, \dot{q}^{\nu} + g_{0 \mu} \, \dot{q}^{\mu}+g_{0 0} \end{equation} I get that: $$\dfrac{\partial T}{\partial\dot{q}^{\lambda}}=g_{\lambda\mu}\dot{q^{\mu}}+g_{o \lambda}$$ where: $\,\, g_{\mu \nu } (q^{\lambda},t) \,\,\,\, g_{\mu 0} (q^{\lambda},t) \,\,\,\, g_{0 0} (q^{\lambda},t) $
Let's assume the indices run from $1$ to $2$ just so that you'll see what is going on. I'll assume that $g_{\mu \nu} = g_{\nu \mu}$. Then
$$ T(q^1, q^2, \dot{q}^1, \dot{q}^2, t) = \frac{1}{2} \left( g_{11} \dot{q}^1 \dot{q}^1 + g_{12} \dot{q}^1 \dot{q}^2 + g_{21} \dot{q}^2 \dot{q}^1 + g_{22} \dot{q}^2 \dot{q}^2 \right) + g_{01} \dot{q}^1 + g_{02} \dot{q}^2 + g_{00} = \frac{1}{2} \left(g_{11} \dot{q}^1 \dot{q}^1 + g_{22} \dot{q}^2 \dot{q}^2 \right) + g_{12} \dot{q}^1 \dot{q}^2 + g_{01} \dot{q}^1 + g_{02} \dot{q}^2 + g_{00} $$
and so, for example,
$$ \frac{\partial T}{\partial \dot{q}^1} = g_{11} \dot{q}^1 + g_{12} \dot{q}^2 + g_{01}. $$