So, +3d6 Initiative character, (Level 3 Wired, Synaptic Booster, or Increased Reflexes), Ambidextrous and Dual Wielding, 5 melee attacks at full pool against 1 to 5 Targets with one Major action / Attack Action? I'm not sure what Melee attacks gain from Multi Attack or Anticipation.
Anticipation is only for ranged weapons. I didn't say that, but I didn't rewrite the whole thing. (However I'd be fine trying it in melee too). If you could use Anticipation in melee then yeah that guy gets up to 5 targets if they're all in melee range with one Major for attack and Minor for Multi attack, and 4 edge to trigger anticipation.
Same character using Ranged attacks, makes 4 FA attacks at - 6 AR, full dice pool, with each Attack action.
Or are the additional attacks from Firing Mode / Melee combat only when Multi Attacking?
Good god yes, only when multi attacking. Sorry apparently that wasn't clear, I haven't written this out before. You still must use the multi attack action to use multiple attacks. And I would take away the FA "bonus" of not having to spend the minor action for multi attack since I'm giving it a single target firing option now and taking away the ability to do multiple small pool attacks against the same target.
An ambidextrous shooter firing FA weapons in each hand could make one single target attack with each weapon (-6AR +3DV) at half pool (full pool with anticipation) against targets anywhere around them (even the same target),
OR
They could make up to four attacks with one weapon (either hand since they're ambidextrous) at -6AR +0DV at half pool (full pool with anticipation) against different targets within X meters of each other. They still can't attack with the off hand if they use the option to multi attack with their gun's firing mode ability.