(Lorentz etc) invariant vector fields (Background: I know some but not much differential geometry, hopefully enough to formulate this post.)
I want to ask about what physicists mean when they say scalar, vector, etc. The answer in differential geometry is something like:
1) Scalar fields are smooth functions on spacetime ($\phi \in C^{\infty}
(\mathcal{M})$)
2) Vector fields are smooth derivations / smooth sections of the tangent bundle ($\vec{A} \in \Gamma(T\mathcal{M})$) 
3) One forms and tensors are defined in the usual way as (multi)linear functions satisfying certain properties.
However when we come to studying physics (maybe in QFT) there seems to be an additional requirement:
4) Suppose the map $\Lambda$ sends spacetime to itself (eg a boost). Then we additionally require:
4a) Scalars transform like:
$$\phi(X) \mapsto \phi'(X') = \phi(\Lambda^{-1}(X)) $$
4b) Vectors transform like:
$$\vec{A}(X) \mapsto \vec{A}'(X') = \Lambda \vec{A}(\Lambda^{-1}(x))$$
... and so on for tensors.
I think the above is not completely correct so my questions are:
Q1) Is this a correct summary of the situation, that a "physical vector field" is a mathematical (differential geometry) vector field plus an invariance condition that is a completely separate condition, or can this be motivated from geometry alone?
Q2) What is the correct formulation of 4? Clearly we don't care about all transformations in $\text{Aut}(\mathcal{M})$ but do we fix a subgroup $G\subset\text{Aut}(\mathcal{M})$ arbitrarily? Usually for QFT etc it would be the Lorentz group I know, but are there situations where different $G$ are considered? Also, my writing of 4(b) doesn't make much sense as it stands unless $\mathcal{M}$ is given a vector space structure which I hope isn't necessary. 
 A: The conditions you've described in #4 are superfluous. If something fits the differential geometry definition of a scalar or a vector, then it's guaranteed that its components expressed in a particular basis transform in a certain way under changes of coordinates.
If you pick a typical manifold with a metric and talk about physics on that manifold, then the manifold does not have anything you can define on it that is a Lorentz boost, nor are there translations, rotations, C, P, or T transformations that you can define. A manifold doesn't even have to be time-orientable.
Nevertheless, any manifold with signature $+---$ is guaranteed to look locally like Minkowski space, so you can define these transformations locally. If something fits the differential geometry definition of a scalar or vector, then it's guaranteed automatically that when you express its components in a certain basis, and then apply one of these transformations, the components change as required. We do want vectors to behave this way under parity, or else we actually wouldn't call them vectors, we would call them pseudovectors or axial vectors.
The source of confusion is probably that you're switching back and forth between two different languages. There is an older language involving components, developed furthest by Schouten, and a newer coordinate-independent language, which can be expressed either in "mathematician notation" as you've been taught or in abstract index notation. It can also be confusing because abstract index notation looks similar to coordinate notation, but abstract index notation is coordinate-free.
