Regarding the definition of connections (covariant derivatives) A connection can be thought of as a map $\Gamma(TM) \times \Gamma(TM) \to \Gamma(TM)$, so $\nabla_X Y$ is section of the tangent bundle. But it can also be thought of as a map $\Gamma(TM) \to \Gamma(T^*M \otimes TM)$, but why is that? I don't see how. 
I thought maybe one keeps $X$ fixed in $\nabla_X Y$ but this doesn't give me a section of $T^*M \otimes TM$. Can someone explain it, please?
 A: A section of $TM \otimes T^\ast M$ is just a $TM$-valued $1$-form. In other words, a section $\sigma \in \Gamma(TM \otimes T^\ast M)$ can be considered as a linear map
$$\tilde{\sigma}: \Gamma(TM) \longrightarrow \Gamma(TM),$$
$$\tilde{\sigma}(p, v) = \sigma_p(v).$$
For a concrete example, consider the constant section $(1,2) \otimes dx \in \Gamma(T\Bbb R^2 \otimes T^\ast \Bbb R^2)$. Then the associated linear map $\Gamma(T\Bbb R^2) \longrightarrow \Gamma(T\Bbb R^2)$ in this case is
$$(p, v) \mapsto ((1,2) \otimes dx)_p(v) = (1,2) \cdot dx_p(v) = (p,(v_1,2v_1)),$$
where $v = (v_1, v_2)$.
If you want to view a connection as a map $\nabla: \Gamma(TM) \longrightarrow \Gamma(TM \otimes T^\ast M)$ as opposed to a map $\nabla_X : \Gamma(TM) \longrightarrow \Gamma(TM)$ for some vector field $X$, then you should imagine it as the map
$$Y \mapsto (X \mapsto \nabla_X Y).$$
In other words, you instead think of $X$ as the argument of the associated linear map
$$\nabla_\bullet Y : \Gamma(TM) \longrightarrow \Gamma(TM).$$
For each choice of $X$ you then get a section $\nabla_X Y \in \Gamma(TM)$, so you should be able to see that $\nabla_\bullet Y$ is a section of $TM \otimes T^\ast M$.
A: It does, if you identify $T^*M \otimes TM$ with $\text{Hom}(TM, TM)$. I'll describe the identification in the linear algebra setting between $V^* \otimes W$ and $Hom(V,W)$. Pick a basis $(v_1, ..., v_n)$ for $V$, take $(\phi^1, ..., \phi^n)$ to be the dual basis of $(v_i)$ for $V^*$ and take $(w_1, ..., w_m)$ to be a basis of $W$. Elements of $V^* \otimes W$ are of the form $\sum a_i^j \phi^i \otimes w_j$. How can we interpret them as maps from $V$ to $W$? Given a vector $v \in V$, the action of this element on $v$ is
$$ \left( \sum a_i^j \phi^i \otimes w_j \right)\left( v \right) = \sum a^j_i \phi_i(v)w_j. $$
If we plug in $v_i$ we see that
$$ \left( \sum a_i^j \phi^i \otimes w_j \right)\left( v_i \right) = \sum_{j} a^j_i w_j. $$
That is, the map defined by $\sum a_i^j \phi^i \otimes w_j$ is just the map represented in the bases $v$ and $w$ by the matrix $(a^j_i)$.
In the other direction, given a map $T : V \rightarrow W$, represent it in the bases $v$ and $w$ by $a^j_i$ and match it to the element $\sum a_i^j \phi^i \otimes w_j$. Just like you expand a vector $w \in W$ in a basis $w_i$ as $w = \sum_j a^j w_j$, where the coefficients are scalars, you expand a map $T$ as $T =  \sum a_i^j \phi^i \otimes w_j $ where the "coefficients" $\sum_{i} a_i^j \phi^i$ are functionals in $V^*$.

As usual, everything passes from the linear algebra setting to the vector bundle setting, and you identify a tensor $\sum a_i^j dx^i \otimes \frac{\partial}{\partial x^j}$ with a map between $TM$ and $TM$, given in the frame $(\frac{\partial}{\partial x^j})$ by the matrix $(a_i^j)$.
