Treewidth and Packing
I can interpret this question in two different ways:

1) When it comes to algorithmic properties of packing problems on graphs of bounded treewidth, Courcelle's Theorem shows that for every fixed $k$ we can optimally solve problems expressible in Monadic Second Order Logic in linear time on graphs of treewidth at most $k$ (see for example http://dx.doi.org/10.1093/comjnl/bxm037 for a survey on the algorithmic properties of bounded-treewidth graphs). As many packing problems can be formulated in MSOL, this proves tractability of many such problems on graphs of bounded treewidth, including Independent Set, Triangle Packing, Cycle Packing, packing vertex/edge disjoint copies of any fixed graph, packing vertex-disjoint minor models of some fixed graph H, and so on. But as this tractability extends to all MSOL-definable problems, it is not specific to packing.

2) When it comes to graph-structural relationships between packings and treewidth, the following might be of interest. Thanks to the work of Robertson and Seymour it is known that there is a function $f \colon \mathbb{N} \to \mathbb{N}$ such that every graph of treewidth at least $f(r)$ contains an $r \times r$ grid as a minor (the original bound for $f$ given by Seymour and Robertson was later improved in collaboration with Thomas; see http://www.sciencedirect.com/science/article/pii/S0095895684710732 for the current-best bound). Hence if you have a structure $S$ such that many copies of $S$ can be packed into an $r \times r$ grid minor, then you know that any graph of large treewidth contains a large packing of copies of $S$. For example, as an $r \times r$ grid (for even $r$) contains $(r/2)^2$ vertex-disjoint cycles, it follows that a graph of treewidth $f(r)$ contains at least $(r/2)^2$ disjoint cycles.