This is totally up to your GM. I'll go over the rules as written to clarify the situation, but Savage Worlds is not a legalistic game and your GM can and should adapt the rules as necessary. (In fact, if you're working with your GM, you might get more favourable rulings than if you spring this on them without warning.)

Rules as written

Normally, fighting with two weapons gives two separate penalties. You get -2 to all actions (which are considered to be simultaneous), and an additional -2 to actions taken with the off hand. So the normal case is a two-gun gunslinger firing both guns with -2 to one and -4 to the other Guns roll.

Ambidexterity removes the penalty for taking an action with an off hand. A gunslinger with Ambidexterity would be making two simultaneous shots, both at Guns -2.

Two-Fisted removes the multi-action penalty when using a weapon (or just fist) in each hand. A gunslinger with Two-Fisted would make two simultaneous shots, one with no penalty and one with -2 for attacking with an off hand.

Combining the two obviously takes care of both penalties, letting you (by a literal reading), attack exactly twice with each of two hands, at no penalty. Because the rules-as-written specifically say "two" and "off hand" (singular), then a literal interpretation of the rules is that anything beyond two is unaffected by these Edges.

Since you can't attack with the same weapon twice ever according to the multi-action rules, your four-armed gunslinger does have some default advantage: you can shoot four times at all, while a two-handed gunslinger can shoot a maximum of twice. Additionally* you get no multi-action penalty for attacking with the additional arms according to the Multiple Limbs racial ability. With Two-Fisted and Ambidextrous you would get the first two without penalty, and the extra two limbs' attacks would have either a -2 (including the off-hand penalty) or no penalty, depending on whether your GM rules that Ambidexterity takes care of one off hand (just the first) or all off hands.

^{* This paragraph edited with corrections to the math to accurately reflect the Multiple Limbs text, as pointed out in comments.}

One way a GM could rule

If it were me GMing, I would totally allow for a four-armed gunslinger to get good with all hands simultaneously. To start I'd say that each pair of arms has a dominant and an off hand, so you already have a slight edge with two hands (not just one) on top of being able to fire four times. For full advantages of Two-Fisted I'd require that it be taken twice – once for the first pair of arms, and again for the other pair of arms. I'd similarly require that Ambidexterity be taken twice to offset the two off hands.

Another GM might just say you can never get that good with all four, and limit you to the rules interpretation that the Edges are limited to "two" literally. Yet another GM might say the Edges cover all your arms just fine, no extra Edges needed. Yet another another GM might rule as I would, but require an Ambidexterity Edge for each off hand.

Talk to your GM and find out how they want to handle it, and remember: if you can do it, NPCs can do it! Your enlightened self-interest should take into account whether you want to face enemies who can easily use four wands at once, for example.

One common case where this might occur is Running and attacking. Running is counted as an action, but since it isn't a Trait roll, there is no –2 penalty. However, attacking is a Trait roll and, since it is one of two actions, it suffers a –2 penalty. A similar situation happens when drawing a weapon and firing in the same round. No roll is needed for drawing, but the attack is at a –2 if done in the same round because it's one of two actions.

In other words, anything a character "does", whether attacking, running or, opening a door, is an action so long as it can't be counted as a free action (e.g. saying a few words, dropping an item, falling prone). Indeed the core rules say that "readying an item, drawing a sword, or other quick tasks usually take one action", and it later defines running as one such action. Opening a door and such seems to fall under this category.

Now what's to prevent taking lots of non-trait roll actions? Mostly the GM saying that it's highly unlikely that the character can run, open a door, eat a sandwich, reload a gun, and write a sonnet all at the same time. The GM can of course throw on some extra rolls if appropriate. For instance, the rules say that if a character tries to draw a gun from a difficult location (such as an ankle holster) and fire in the same turn, they should make an Agility (–2) check because they are trying to pull off two difficult tasks in a hurry. I could see the same thing happening if the character tried to do a lot of noncombat actions in one turn: the GM might give an additional Agility roll or another appropriate roll to pull it off. With a failure, the character just klutzes around.