Do rockets or destroyed systems count for the green d8?
If I make a melee mech with all of its red dice devoted to a claw attack (hand-to-hand range), can I put a rocket on my mech and still get the green d8 movement bonus?


Yes.  Since Single Shot Rockets don't count as systems, the bonus Green d8 for having no ranged weapons ignores them.


  Even when I'm firing my rocket?


Yes.  (See above.)


  If my mech had a gun, but it got blown off so now it doesn't anymore, do I get the green d8 movement bonus for my last big dramatic charge?


Yes.  Once you have no ranged weapon systems left you gain the bonus Green d8.