Coordinates that cancel the 64 affine connection components at a point

Both questions are easily answered once we realize that $partial_mu a^lambda_{ nu}=partial_mu partial_nu tilde{x}^lambda = partial_nu partial_mu tilde{x}^lambda = partial_nu a^lambda_{ mu}$.

1. The antisymmetric part of $Gamma^{lambda}_{mu nu}$, which we denote as $Gamma^lambda_{[munu]}$ (where $[mu,nu]equiv frac{1}{2}(munu-numu)$), transforms as a tensor. The part of the transformation of $Gamma^{lambda}_{munu}$ which does not transform as a tensor is symmetric under the exchange of $mu$ and $nu$, which can be shown using the property above. Therefore, it is not possible to find coordinates where $Gamma^lambda_{munu}=0$ in general. If it were possible, then $Gamma^lambda_{[munu]}$ would vanish, and then by properties of tensors, $Gamma^lambda_{[munu]}=0$ would be zero in every coordinate system. Indeed, the antisymmetric part of $Gamma^lambda_{mu nu}$ is a way to measure the torsion -- so coordinate invariance by itself is not enough to set $Gamma^lambda_{[munu]}=0$, you need an additional assumption that the torsion vanishes. This assumption is made in GR, but logically speaking this is an extra assumption you need to make.

2. As we established above, $partial_mu a^lambda_{ nu}$ is symmetric under exchanging $mu$ and $nu$. It therefore has 40 components (4 possibilities $lambda$ times 10 possibilities for the symmetric pair ($mu,nu$). This gives enough freedom to cancel the freedom in the symmetric part of $Gamma^{lambda}_{mu nu}$.

The OP is not the only person who has wondered about this. And even a textbook can be mistaken, or at least misleading, about this.

I have in my possession a textbook which assigns this as an exercise. It says that for any point $P$ of a manifold and any connexion on any neighbourhood of it $U$, there exist coordinates (in a neighbourhood of $P$ which is possibly smaller than $U$) such that all $Gamma_{ij}^k$ are zero at $P$. Part of the exercise is to construct the coordinates using the geodesics of $Gamma$ which begin at $P$.

Not every textbook is error-free, and Doubrovine, Novikov, and Fomenko, is certainly not. But there it is, on p. 285, of Geometrie Contemporaine, Methodes et Applications, Premier Partie, Geometrie des surfaces, des groupes de transformations et des champs, Moscow, 1982, tranlated from the Russian edition of 1979. I understand that the new revised Russian edition is freely available on the web. Also, I have read this theorem elsewhere for the special case of connexions which are torsion free and compatible with the metric. But neither of those assumptions are made here.

One can not trust one exercise posed in one textbook, and Moretti has solved their exercise under the hypothesis that there is no torsion.

So the same question that the OP wanted clarified is actually the subject of a mistake in a reputable textbook.

If the connection has torsion, the answer is negative. Indeed, if you found a coordinate system around a point where all connection coefficients vanish at that point, then also their antisymmetric part would vanish, and this is not possible because the torsion is a tensor so that, if it vanishes in a coordinate system, then it has to vanish in every coordinate system.

Hence, let us assume that the connection is symmetric without assuming that it is metric. What follows does not need a metric, just an affine symmetric connection on a manifold with generic dimension $n$.

Consider a point $p$. In that case you can define the local diffeomorphism known as the exponential map at $p$. The map identifies an open neighborhood of the tangent space at $p$ with a neighborhood of $p$.

$$exp_p{v} = gamma(1,p, v)$$ Where $t mapsto gamma(t,p,v)$ is the geodesic with initial point $p$ and initial tangent vector $v$.

(Actually there are a few mathematical subtleties. First, one has to prove that it is always possible to include $1$ in every maximal parameter domain of all geodetics emanated from $p$ if $v$ stays in a suitable small neighborhood of the origin of $T_pM$. Secondly, one has to prove that the map written above is in fact a diffeomorphism, i.e., smooth, bijective with inverse smooth, by possibly shrinking the said neighborhood. All that can be fixed with a little effort.)

Fixing a basis in the tangent space at $p$, the components of $v$ are coordinates of the corresponding point on the manifold. Such a coordinate system is known as a normal coordinates system centered on $p$.

The geodesics emanated by $p$ are straight lines exiting the origin in that coordinate system, $$v^a(s) = sv_0^atag{1}$$ This is an immediate consequence of the identity arising from the geodesic equation (and the uniqueness theorem for that differential equation) $$gamma(s,p,v_0) = gamma(1,p, sv_0)$$ Eventually, (1) easily implies from the geodesic equation that the connection coefficients must vanish at $p$ in the said coordinate system. In fact, it must hold at $s=0$ $$Gamma(0)_{ab}^cv^av^b=- frac{d^2sv^c}{dt^2}|_0=0$$ for every choice of the coefficients $v^a$. Using the fact that the connection is symmetric, replacing $v$ for $upm w$, you find $$Gamma(0)_{ab}^c=0.$$

If the connection is metric (Levi-Civita), we also have for free that $$frac{partial g_{ab}}{partial x^c}|_{p}=0$$ in the said coordinate system. It arises by just writing the above derivative in terms of connection coefficients (using the metricity condition).

ADDENDUM. Actually the exponential map is defined also when the connection has torsion (I did not use that hypothesis in defining the exponential map). In that case, the reasoning above proves that the symmetric part of the connection coefficients vanishes at the center of a normal system of coordinates. The antisymmetric part cannot be cancelled in any cases as said at the beginning of this answer.

Also, to close the loop, let's prove this by controposition. Let's assume that, for some torsion-having metric, we HAVE produced a transformation that makes all $Gamma$ zero at some point. Then, at this point, we have:

$$nabla_{a}v^{b} = partial_{a}v^{b}$$

Now, transform to some other coordinate system. This will require that we multiply $v^{b}$ by the transformation matrix $P^{a}{}_{b} = frac{partial y^{a}}{partial{x^{b}}}$

Now, we have, in this new coordinate system (suppressing multiplication on the left by two factors of $P$ on both sides to transform the basis components:

$$nabla_{a}v^{b} = partial_{a}left(P^{b}{}_{c}v^{c}right) = P^{b}{}_{c}partial_{a}v^{c} + v^{c}partial_{a}P^{b}{}_{c}$$

The first term becomes just the partial derivative term again, because the factors of P that we suppressed satisfy the identity $P^{a}_{b}P^{b}{}_{c} = P^{a}{}_{c}$ by the chain rule. That second term must be the Christoffel symbol in our new frame with nonzero Christoffel symbols, so:

$$Gamma_{ab}{}^{c} = partial_{a}P^{c}{}_{b} = frac{partial^{2}y^{c}}{partial x^{a} partial x^{b}}$$

This cannot have any torsion associated with it, because it is symmetric in a and b.