You can walk or run as part of a Rush in to a brawl. Any movement type may be used to rush an enemy; to move one character into base-to-base contact with another. Otherwise you have to keep moving characters 1" apart...
There is no penalty to your Fighting skill if you run into a brawl, think of it as a sort of 'charge bonus'.
A character can not run and then shoot, with a -1d penalty for having run. But If shot at, that character could REACT as a defender, by firing back at a -1d penalty for having run during its activation. Multiple -1d penalties would apply if you ran, shot and then fought as a reaction.
*Edit* I got the bit about running & shooting wrong. I've corrected it just in case it pops up as a search result...