Is there something special that needs to be done with "heavy" weapons or hand to hand?
Heavy Weapons have a 'restricted' use, as you can't use them for Overwatch Attacks (p. 27). Hand to Hand weapons can be used for out-of-path attacks, as long as you're attacking an adjacent space.
Also, is there a minimum range for each type of weapon (not a max range)?
There's no minimum range. If you're firing at something on your path, you can hit it. Some Mental Attacks or Special Items do have range restrictions (example: Karl Zermann's Doom can attack over a distance of Current Mental Value + 1 regardless of the path your target is on.
One last thing: if a player has a certain number of dead characters, does that affect the initiative roll at all?
No, unless they carried equipment that gave you a bonus to the initiative roll. You'd lose said bonus in that case (example: Generalleutnant in Hermann Von Heïzinger's Command Pack).