You don't need to worry about buffers or Polygons.
I think the tool you are looking for is called "Dissolve" in the Data Management/Generalization Toolbox. You have to decide whether to use the parameter DISSOLVE_LINE or UNSPLIT_LINES - the latter will only create a single line for contiguous features - i.e. lines that share an endpoint.
Dissolve and its usage, in Arcpy, is thus:
Dissolve Data Management
It effectively aggregates your geographies on specified attributes i.e. in your case, it's name.
The Unsplit lines parameter with two options, DISSOLVE_LINES and UNSPLIT_LINES, only applies to line input. When the default DISSOLVE_LINES option is specified, lines are dissolved into a single feature. When UNSPLIT_LINES is specified, only two lines that have a common endpoint (known as pseudonode) are merged into one continuous line.
I should add that it is very simple to use, either as a standalone script, or as a command in the python window in 10.
EDIT: You might want to have a look at the answer here if you're prepared to use another software package. From what you've told us it looks like arc may not be the best tool to work with your particular dataset.