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.

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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.