If we use the notation $(TM, p_M, M)$ for the tangent bundle of any manifold $M$, then you are right to think that $T^{\ k}M$ has $k$ natural vector bundle structures over $T^{\ k-1}M$ and so on down to $M$, making a diagram which is a $k$-dimensional cube. Such a structure is a $k$-fold vector bundle (See articles by Kirill Mackenzie) and $T^{\ k} M$ is a particularly symmetrical one. An easy way of writing the $k$ bundle maps would be
$$T^{\ a}(p_{T^{\ b} M}) $$
for $a+b+1 = k$, where this means that we are taking the $a$-th derivative of the tangent bundle projection $T\ (T^{\ b}M)\to T^{\ b}M$, yielding a map now from $T^{\ k}$ to $T^{\ k-1}$.
To see the symmetries of $T^{\ k} M$ it is more convenient to describe the functor in a direct way rather than as a $k$-fold composition -- just as you might think of tangent vectors as infinitesimal curves, you can think of points in $T^{\ k} M$ as infinitesimal maps of a unit $k$-cube into $M$. The restrictions to the $k$ faces of the cube (going through the origin) gives your $k$ maps.
From that point of view, it is clear that you can permute the $k$ coordinate axes and get another map of a cube into $M$, so that the functor $T^{\ k} M$ has a $S_k$ group of natural-automorphisms.
Incidentally, to make the above into a definition of $T^{\ k} $, you could do the following:
Consider the fat point $fp$, which you should think of as a space whose smooth functions form the ring $\mathbb{R}[x]/(x^2)$. Then the tangent bundle $TM$ can be thought of as the space of maps from the fat point to $M$, i.e. $TM=C^\infty(fp,M)$. Such maps, by the way, are just algebra homomorphisms from the algebra $C^\infty(M,R)$ to $\mathbb{R}[x]/(x^2)$. You can check that such a map has two components $f_0 + f_1 x$, and that $f_0$ is a homomorphism to $\mathbb{R}$ defining a maximal ideal (i.e. a point $p$ in $M$) and $f_1$ defines a derivation (i.e. a vector at $p$).
In precisely the same way you can consider a cubical fat $k$-point $kfp$ with functions
$$\mathbb{R}[x_1,...,x_k]/(x_1^2,...,x_k^2)$$
and then define $T^{\ k} M = C^\infty(kfp,M)$. Then you can see the symmetries as automorphisms of the above algebra, and the $k$ maps you ask about as homomorphisms $C^\infty(kfp)\to C^\infty\big((k-1)fp\big).$
There are probably more subtle things to be said about these higher iterated bundles but I hope the above is at least correct.
Yes for "hope" 1. This theorem was proven by Moser using volume-preserving flows. A manifold with a volume form is the same thing as a manifold with an atlas of charts modeled on the volume-preserving diffeomorphism pseudogroup. He found an argument that can be adapted to either the symplectic case or the volume case. I cited this result in my paper A volume-preserving counterexample to the Seifert conjecture (Comment. Math. Helv. 71 (1996), no. 1, 70-97), where I also established a similar result for the volume-preserving PL pseudogroup. In the PL case, the corresponding decoration on the manifold is a piecewise constant volume form.
In my opinion, the most exciting result on this theme is the Ulam-Oxtoby theorem. (But Moser's version is the most useful one and the most elegant.) The theorem is that if you have a topological manifold with any Borel measure that has no atoms and no bald spots, then it is modeled on the pseudogroup of volume-preseserving homeomorphisms. For example, you can start with Lebesgue measure in the plane and add uniform measure on a circle, and there is a homeomorphism that takes that measure to Lebesgue measure.
For a long time I have wondered about the pseudogroup of volume-preserving Lipschitz maps. The question is whether there is a corresponding cone of measures, and if so, how to characterize it.
Best Answer
Your answer 1. is local and coordinate dependent, thus 2. is not well defined. For a chart change (a diffeomorphism) $\phi$ the mapping $TT\phi$ looks as follows: $$ (TT\phi)(x,dx,\delta x,d\delta x) = (\phi(x),\phi'(x)(dx),\phi'(x)(\delta x), \phi''(x)(d x,\delta x) + \phi'(x)(d\delta x)). $$ Note that the forth component transforms exactly as Christoffel symbols for a linear connection do.
For a description of the second tangent bundle of a manifold see 8.12 and 8.13 of here. For a comprehensive description of higher order order tangent bundles etc, viewed as product preserving functors on the category of smooth manifolds, see chapter VIII of here, or the original paper. Thsi description uses Weil algebras and is related to what you are trying to say at the end of your question.