First a word of warning: I am not a trained algebraic geometer, so it is possible (likely) that these questions are inappropriate for MO, if so: my apologies.

Said this: As far as I understand the tangent bundle $TX$ of a scheme $X$ is defined as the spectrum of the symmetric algebra of its sheaf of differentials, and so (I guess) pointwise nothing else than the associated scheme to the Zariski tangent space. The "projection map" $\pi: TX \rightarrow X$, which is a morphism of schemes, gives a local description of the tangent bundle, by $TU \cong \pi^{-1}(U)$, where $U$ is an affine open set in $X$.

My (first) question is the following: (1) is there a "global" description of the vector space structure (which I suppose exists?!) on the tangent bundle of a scheme. I mean addition should somehow be defined by a scheme morphism $TX \times_X TX \rightarrow TX$, where $TX \times_X TX$ is the scheme fibre product, and scalar multiplication by a scheme morphism $\mathbb{C} \times TX \rightarrow TX$. Does there exist such a description? Or what other basic properties does the tangent bundle of scheme have? (A reference would also be appreciated)

As a second question: (2) Is the total space associated to the tangent sheaf (defined as the $\mathcal{O}_X$ dual of the cotangent sheaf, so the total space associated to a quasicoherent sheaf) the same as the tangent bundle? I guess there could be some issues with "taking the dual of sheaves"?!