I've been trying to understand the decomposition of the second tangent bundle, $TTQ$, of a smooth Riemannian manifold, $(Q,g)$, into so-called horizontal and vertical bundles, $TTQ = HTQ \oplus VTQ$, using the Levi-Civita connection $\nabla$. I know that such a decomposition can be applied to much more general vector bundles over $Q$ and that $TQ$ is just one of the simplest cases. But even this simple case is confusing for me. So, I want to consider the super simple case that $Q$ is replaced by some $n$-dim affine space, $X$, with associated vector space, $(\mathbf{U},g)$, where $g\in\mathbf{U}^*\otimes\mathbf{U}^*$ is some inner product/metric tensor (I'm thinking of these as the affine/vector space views of Euclidean or Minkowski space, but that need not be the case). Let $u_x=(x,\pmb{u})\in TX$ be some point in the tangent bundle. Using $T_{x}X \cong \mathbf{U}$, I make the following identifications (please tell me if these are wrong):
$$ T_{x}X = \mathbf{U} \quad,\qquad TX = X \times \mathbf{U} \quad,\quad T_{u_x}(TX) = \mathbf{U}\oplus\mathbf{U} \quad,\quad T(TX) = (X\times\mathbf{U})\times (\mathbf{U}\oplus\mathbf{U}) $$
$$ T_{u_x}^*(TX) = \mathbf{U}^*\oplus\mathbf{U}^* \quad,\quad T^*(TX) = (X\times\mathbf{U})\times (\mathbf{U}^*\oplus\mathbf{U}^*) $$
where $\mathbf{U}^*$ is the dual space and, above, I'm using $\oplus$ only for the direct sum of vector spaces to keep the point-vs-vector distinction clear. If the above is correct, then:
Question 1. are the horizontal and vertical spaces over $u_x\in TX$ as follows?:
$$ H_{u_x}(TX) = \mathbf{U} \oplus \{\mathbf{0}\} \quad,\quad V_{u_x}(TX) = \ \{\mathbf{0}\} \oplus \mathbf{U} \quad,\quad H_{u_x}^*(TX) = \mathbf{U}^* \oplus \{\mathbf{0}\} \quad,\quad V_{u_x}^*(TX) = \ \{\mathbf{0}\} \oplus \mathbf{U}^* $$
whith the corresponsing bundles then being, for example, $V(TX) \cong (X\times\mathbf{U})\times ( \{\mathbf{0}\} \oplus \mathbf{U} )$, and similarly for the others?
Question 2. If the above is correct, are the horizontal and vertical lifts of $\pmb{u}\in T_x X =\mathbf{U}$ given simply by:
$$ \pmb{u}^{\mathrm{H}} = \pmb{u} \oplus \mathbf{0} \in H_{u_x}(TX) \qquad,\qquad \pmb{u}^{\mathrm{V}} = \mathbf{0} \oplus \pmb{u} \in V_{u_x}(TX) $$
and are the lifts of $\pmb{u}$ to the horizontal/vertical cotangent spaces (using the same superscripts) simply:
$$ \pmb{u}^{\mathrm{H}} = g^{\flat}(\pmb{u}) \oplus \mathbf{0} \in H_{u_x}^*(TX) \qquad,\qquad \pmb{u}^{\mathrm{V}} = \mathbf{0} \oplus g^{\flat}(\pmb{u}) \in V_{u_x}^*(TX) $$
where $g^{\flat}(\pmb{u}) = g\cdot\pmb{u} \in \mathbf{U}^*$ is the usual inner product isomorphism, $g^{\flat}:(T\!\pmb{.}\,X=\mathbf{U})\to (T\!\pmb{.}^*X=\mathbf{U}^*)$.
context: I'm interested in the application of this in classical mechanics where $Q$ is an embedded Riemannian submanifold of $X=E^N$ and thus $TTQ$, $HTQ$, and $VTQ$ are all embedded in $TTE^N$. I have been doing some things operating under the assumption that what I have said above is true but I am not at all confident about it. Everything I've found about vertical/horizontal bundles seems to be written for people much better at math than myself. I'm not looking for a detailed definition of horizontal/vertical bundles. I can regurgitate the proper definitions, I just don't get what they mean. Knowing if I am right/wrong about all the above will help greatly.