Is there a way to pullback affine connections to get again affine connection?
The standard pullback construction you describe gives a connection $f^* \nabla$ on the bundle $f^* TN$, so the question is how to naturally turn this in to a connection on $TM$ using the bundle map $Df : TM \to f^* TN.$ More abstractly, we have two bundles $E \to M, F \to M$ and a bundle map $\phi :E \to F$, and we want to know what conditions on $\phi$ are required in order to pull back a connection $\nabla$ on $F$ to a connection $D$ on $E$.
In order to get a feel for what this "pullback" should do, we can first assume that $\phi$ is an isomorphism (which corresponds in the original problem to $f$ being a local diffeomorphism). In this case we can use $\phi$ to identify $E$ and $F$, and so all structures on $F$ transfer naturally to $E$ (and vice versa). In particular we have the formula $$ D_X \xi = \phi^{-1} \nabla_X (\phi(\xi))$$ for a section $\xi \in \Gamma(E).$
Now we can try to relax our assumptions: what do we really need to assume about $\phi$ in order for this formula to define a connection on $E$? Well, immediately we see that for $\phi^{-1}$ to make sense we need $\phi$ to be injective. But something else can go wrong: we need $\nabla_X(\phi(\xi))$ to be in the image of $\phi$, but connections are surjective on fibers; so in order for this to make sense we need to assume that $\phi$ is surjective. Thus we can't gain anything if we want to stick strictly to this formula: we need $\phi$ to be an isomorphism.
To reconcile this with the induced connection in Riemannian geometry, remember that we use extra structure there: the induced connection is the tangential projection of the pullback connection. That is, for a Riemannian immersion $f : M \to N$ with corresponding injective $\phi = Df : TM \to f^*TN,$ we have $D_X Y = \pi (\nabla_X \phi(Y))$ where $\pi : f^*TN \to TM$ is the left-inverse of $\phi$ given by orthogonal projection on to $TM$. Without the Riemannian structure to single out the correct choice of $\pi$, we have no way of distinguishing between the many left-inverses of $\phi$ and thus no canonical induced connection.
What is the relation between Christoffels symbols for the connection and for its pullback?
If we fix local coordinates $x^i$ on $M$, $y^\alpha$ on $N$ and a local frame $E_A$ for a vector bundle $E \to N$, then the pullback bundle $f^* E$ has a local frame given by $E_A \circ f$, and the corresponding connection coefficients defined by $$^{f^* \nabla}\Gamma^B_{Ai}(E_B \circ f)=(f^* \nabla)_{\partial/\partial x^i}(E_A \circ f),\;\;\;^{\nabla}\Gamma^{B}_{A\alpha} E_B=\nabla_{\partial/\partial y^\alpha} E_A$$ are related simply by $$^{f^* \nabla}\Gamma^B_{Ai} = {}^\nabla \Gamma^B_{A \alpha} (df)^\alpha_i.$$
