I am 90% certain of the response to this, but Let me stay positive:

Automatically (e.g. not while using 301 modifier), does mod_rewrite route internally within Apache or redirect the customer via http headers as well as other method?

All my intuition, research, and experience signifies the redirection is performed internally. By 'internally', I am talking about the client is oblivious that mod_rewrite is being used. For instance, think about the following rule:

RewriteRule ^([^/]+)$ dispatcher.html?cat=$1

Whenever a request is perfect for which this rule is applicable (e.g. example.com/testing), the request is rerouted (e.g. to example.com/dispatcher.html?cat=testing). My knowledge of mod_rewrite would be that the module simply rewrites the request therefore it seems the original request found example.com/dispatcher.html?cat=testing.

Is correct?

Automatically (e.g. not while using 301 modifier), does mod_rewrite route internally within apache or redirect the customer via http headers as well as other method?

The example you show is going to be rewritten internally.

Should you clearly pressure a complete URL, a header redirect will occur except (basically browse the docs right) when the full URL indicates exactly the same domain out of the box presently being processed, by which situation the part indicating the server is going to be removed, as well as an internal redirect performed.

This list in the paperwork shows all possible situations:

Given Rule                                      Resulting Substitution
----------------------------------------------  ----------------------------------
^/somepath(.*) otherpath$1                      invalid, not supported

^/somepath(.*) otherpath$1  [R]                 invalid, not supported

^/somepath(.*) otherpath$1  [P]                 invalid, not supported
----------------------------------------------  ----------------------------------
^/somepath(.*) /otherpath$1                     /otherpath/pathinfo

^/somepath(.*) /otherpath$1 [R]                 http://thishost/otherpath/pathinfo
                                                via external redirection

^/somepath(.*) /otherpath$1 [P]                 doesn't make sense, not supported
----------------------------------------------  ----------------------------------
^/somepath(.*) http://thishost/otherpath$1      /otherpath/pathinfo

^/somepath(.*) http://thishost/otherpath$1 [R]  http://thishost/otherpath/pathinfo
                                                via external redirection

^/somepath(.*) http://thishost/otherpath$1 [P]  doesn't make sense, not supported
----------------------------------------------  ----------------------------------
^/somepath(.*) http://otherhost/otherpath$1     http://otherhost/otherpath/pathinfo
                                                via external redirection

^/somepath(.*) http://otherhost/otherpath$1 [R] http://otherhost/otherpath/pathinfo
                                                via external redirection
                                                (the [R] flag is redundant)

^/somepath(.*) http://otherhost/otherpath$1 [P] http://otherhost/otherpath/pathinfo
                                                via internal proxy