Does the DOM do that for me?
The DOM by itself (pls consult http://www.w3.org/DOM for details) “does” nothing for you; it’s an API specification.
DOM implementations however, e.g. libxml2, can assist you on processing XML-formatted data, e.g. COLLADA instance documents.
In your case, the requirement appears to be loading data from COLLADA documents or strings into internal data structures of your application.
Vector- and Facedata for meshes, e.g., would reside in so-called “Text Sections” you could e.g. use as strings in python, C/C++ or other languages and tokenize and covert to float or integer binary data which you could fill into and manage in data structures internal to you app.
For the “COLLADA DOM” the following definition is found e.g. in the file “Readme.txt” of the COLLADA DOM package
The COLLADA DOM is a set of libraries for loading and saving COLLADA documents that can contain 2D, 3D, physics and other types of content.
Thus, this contains the implementation of a higher level and more specialized interface for accessing XML data of COLLADA instance documents than general XML libs like e.g. libxml2 do.