Very good questions, that many people will ask themselves. I will answer your questions using the Savage Worlds Deluxe rulebook from 2011 (not SW:EX from 2007), in case it helps anyone:
Tonight in my Savage Worlds (...) game, several of the
bad guys, all Extras, were shot, took damage exceeding their
Toughness, and were put out of the fight--which I believe is
As someone already said above, exceeding their toughness only makes them shaken. Exceeding their toughness with a raise, or shaking them a second time with a physical wound would incapacitate them and put them out of the game.
These same guys made their Vigor rolls, per the Aftermath section
(...) Does this mean they are considered Shaken and suffering from
Wounds? If they are wounded, how many wounds? What penalty to trait
If they pass their Vigor roll in the aftermath of the fight they remain incapacitated (p78), i.e. they are alive but cannot do anything apart from free actions like talking. They cannot even walk unless they succeed on a second Vigor roll ("walking wounded", p78). They must be left in the place or carried away with care.
If they fail the Vigor roll, they are dead (p78). End of story.
If they succeed the Vigor roll with a raise their wounds are only superficial and they are no longer incapacitated (p78). It is assumed that they lost conscience but woke up after the fight, or maybe got a minor injury and shied away from battle (cowards!). In any case, they are perfectly functional ("as new").
Extras do not have wounds: they are either dead, incapacitated, or perfectly fine (p68).
One of them was healed by a PC's Ally. The healing roll was sufficient
to heal one wound. How many wounds does he have now? What penalties?
Is he still Shaken?
If an extra is healed by the Healing skill or the Power, the extra becomes completely functional ("as new"). Extras do not have wounds: they are either dead, incapacitated, or perfectly fine (p68). A success in Healing is enough to heal an incapacitated extra.
If the extra is healed in the middle of a combat (using the Healing power) the extra comes back to life functional (no penalties) but shaken (p114). Also note that before you can use the Healing power on an extra while in combat you must determine whether the extra is dead (Vigor roll, as per the aftermath rules in p78). If the extra is dead, no healing is possible (p114).
If the extra is healed after the combat has finished then they are not shaken (it is assumed that they repeat their Spirit roll every few seconds until they succeed).
Note: You cannot use the Healing skill while in combat. The skill takes 10 minutes per attempt, and the combat is over way before that.
In the fiction, is he still suffering from a bullet wound?
If they passed the Vigor roll and they got cured, it is assumed that the bullet wound was bad but not so bad that it could not be cured. Maybe it was just a scratch on the forehead that covered their face with blood (really messy if it ever happens to you!) but after cleaning and a few stitches they are ready for action.
Lastly, they all got Gut Shot on the Injury table. Does this mean that
in the fiction they are seriously wounded, but somehow able to manage?
In the Savage Worlds Deluxe rulebook it is very clear that only Wild Cards roll on the Injury Table (p68). Extras never roll on the Injury Table according to the rules (but you are free to do it as GM, of course, if it fits your style / story).