I think the word "curvature" causes a bit of confusion to you.
The curvature of a connection is something related a the connection you put on a manifold that in principle do not depend on the metric you have put on the manifold.
If you want to rediscover the intuitive concept of curvature (Gaussian curvature* for example) used in the study of surfaces embedded in $\mathbb{R}^3$ (which is a metric concept), you start with a metric $g$ and take the curvature tensor $R$ **of the induced Levi-Civita connection** of $g$.
In this case $R$ contains all the geometric information you expect. For example from $R$, you can construct the Sectional curvature $K$.
$K(X,Y)$ will give you the Gaussian curvature of an embedded surface generated geodesically by the vectors $X,Y$. It is a very geometric interpretation.
Also the Ricci tensor is a sum of sectional curvatures and the Scalar curvature is another sum of sectional curvatures.
So the curvature tensor $R$ will give you really geometric informations coming from the metric but you have to consider the L-C connection.

Consider for example $(S^2,g_1)$ and $(S^2,g_2)$ Riemannian where $g_1$ is the metric given by the standard embedding in $\mathbb{R^3}$ with the standard flat metric and $g_2$ is the metric pulled back from an ellipsoid embedded in $\mathbb{R^3}$. You have two L-C connections, $C_1$ and $C_2$.
If you compute the curvature tensor for $C_2$ (or another random connection not related to $g_1$) it will not give the intuitive geometric informations relevant for $(S^2,g_1)$ wich is (iso)metrically a sphere, even if you consider the scalar curvature.

2) No, because if you parallel transport a two vectors along a curve using a connection not even **compatible** with the metric (compatible means that $\nabla_X \langle Y,Z\rangle = \langle \nabla_X Y , Z \rangle + \langle Y , \nabla_X Z \rangle$ notice also that the L-C connection is by definition compatible with the metric), you can get different angles.

Instead if you use a connection compatible with the metric the parallel transport preserves the scalar product (given by the metric).

*I do not know if you are familiar with the Gaussian curvature of surfaces embedded in $\mathbb{R}^3$ but it is a fantastic scalar that for example if positive tells that the surface locally resembles a sphere, if negative instead tells that it resembles a saddle (like the saddle of a torus for example), the cylinder instead and the plane have Gaussian curvature 0. Probably it is really what fit best the layman term "curvature".

The Gaussian curvature is a metric concept (because it depends only on the first fundamental form) and is nothing more than the Sectional curvature of the Levi-Civita connection associated to the metric obtained restricting the standard flat metric of $\mathbb{R}^3$ to the surface.
If you for example give *another* connection to the surface and compute the sectional curvature of it, it would be like computing the Gaussian curvature of *another* embedding (well, provided that the connection you are using is the L-C of some other metric).

**Answers to the comments**

1) I don't know any geometric interpretation in this case. The only thing I can think is that the curvature is linked (as you pointed out) to the holonomy i.e. what happens when you parallel transport a vector along a loop. In general if the curvature tensor is not trivial then the vector can be different.

2) The fact that the vector can be different can be stated for example in terms of angles onece you consider an auxiliary Riemannian metric. But this is just to state it in terms of angles. The point is that the vector after a loop is different.

2b) they will not result in the same change of angles, the two Riemannian metrics could measure different angles, if the transported vector is different they will measure it but the effective angle measured can be different. (Notice that this is not the case I was dealing with when talking about compatibility, in that case I was considering the parallel transport of two vectors and studying how the angle between them changes, in this case we are studying a single vector and after a loop we see how it change)

Maybe you should take a look at some books about Holonomy groups, unfortunately I know almost nothing about them.