How did they bring Fischer back to life within the 3rd layer of the dream?

Fischer is shot in the 3rd layer of the dream and enters the limbo.

In limbo, you are actually in your own mind, you don't follow the rules of others. What is more, you are so "sedated" (so deep in your subconscious) that you forget reality. This means, you don't have contact to anything else. You are numb (like in a coma). If you pay attention to what the limbo looks like, you'll see that being in the collapsing building (level 3) or in the water (level 1) doesn't seem to affect Cobb.

Probably that is why Fischer is fine. This is also the case of Saito. When Fischer is rescued, he gets very fast from limbo to dream level 1 (because of the sychronized kicks) and, as we remember, he was fine there. So you might say that at resurrection he "jumps over" two layers and returns to where he was just fine.

So what happened to Saito? Well, he and Cobb are in limbo and miss those synchronized kicks. The others get the kicks and wake up, destroying the levels 2 and 3. That means we have now only level 1 and limbo. So there are 2 possibilities:

1. Saito wakes from limbo directly to reality because they stay longer in limbo. He is not affected psychically because Cobb is there to remind him of the reality and to make him realize where he actually was (this was the danger: not to know what's real anymore).

2. They wake up in level 1 and manage to heal Saito (Fischer finds peace and has something to meditate, so his projections will not be a threat anymore). In level 1, Saito was shot in the stomach, giving him about 45 minutes before he died (so I have read). So after 15 minutes of driving, he still has 30 minutes, enough to create a hospital or an impossible structure to get fast to a doctor. I think he could be saved.

Now, you also asked why they didn't let Saito get into limbo from the beginning. Well, that was not a great plan, since going there is a great risk (you might forget who and where you are) so going after him would not be such a great idea, unless they had no other choice (which happens in the 3rd level).

Unfortunately I cannot answer this like the other questions on Inception in a positive manner.

This appears to be a gaping hole in the plot. Ariadne, the second least experienced of the team proposes to follow Fischer down into Limbo, in order to rescue/retrieve Fischer. She and Cobb do this and both Ariadne and Fischer are brought back to the 3rd dream layer by falling off a collapsing building, a combination of the falling / dying mechanism for going up a dream layer. Since if Ariadne was wrong, she could end up in Limbo for years - she seems strangely confident and unafraid to try this.

As you say, if that was possible why did they not let Saito die and do the same thing to him - follow him down there, give him a kick or kill him to push him back up to Dream Level 1 magically healed.

I can only surmise that Ariadne and Cobb were desperate and willing to try anything. This is surprising from Ariadne as she had little to loose from giving up at this point, but not surprising from Cobb. Then it would appear that whilst they are sedated too heavily in the real world for waking up when dying in the dream, the dying/kick mechanism for moving up dream layers is not disabled by the sedative even in Limbo. This may have been a calculated gamble on their part.

It could have something to do with Mr. Saito getting injured in the first layer. Maybe the effects of going in and out of limbo depend on the layer of the dream that the person is in.

I'm just speculating here, but I'm guessing that Fischer was "revived" because he re-entered the dream. Limbo doesn't seem to be a layer, but an overall "area" that people can get to in several ways: dying in a dream, going through enough dream layers, etc., so it's possible he just came back in to the third dream layer when he got killed in limbo.

They probably couldn't help Mr. Saito because he was dying in another dream layer. He wasn't necessarily affected physically in the following dream layers except for his body's reactions to the shot from the first layer, so they couldn't really heal something that happened in "reality" (the first dream layer)

It's dreaming. We don't know much about it, so there aren't a lot of rules to it in Movie Land.

Layer 3 is Eames' dream from level 2, and he can dream what he wants. If a dreamer can create buildings, and defy physics, why not also being able of resurrecting one of the co-dreamers. His needing of the use of a defibrillator could indicate that he is not a thát almighty and conscious dreamer, but I think it is better movie than when he would have revived Fischer by clapping his fingers.

Saito staying as wounded from level 1 just suggests Yusuf has some more dreaming experience to acquire.

Reference : Inception Explained

To talk of what happened, Mal shoots Fischer.
Mal is a Cobb's projection.

Cobb's mind is very affected by her death, he holds himself responsible for it. When his mind projects Mal, he projects her in a way that is out to ruin it for Cobb, that is his out-of-control guilt doing that.

Cobb, therefore, projects a Mal that attempts to kill Fischer. Her gun and the bullet that hits Fischer is also projected by Cobb. All the other people protecting the fortress is Fischer's projection.

Fischer experiences being shot in the 3rd level of the dream. The mind tends to believe what it thinks it sees and thinks it feels. Fischer starts to die. Once he dies he will wash up on the shores of Limbo. But that hasn't happened yet. Cobb and Ariadne think there is still one shot to get Fischer back. The way to do it by going down in to Cobb's mind at this point because, after all, Fischer is dying because of Cobb's mind. Fischer's dying mind can be reached using Cobb's mind.

They hook up the dying body of Fischer and Ariadne to Cobb's mind and go one level further down. Now, they do reach limbo, but this is because only 3 levels of dreams were architected. Neither did Cobb nor Ariadne die and reach Limbo. This is why they know where they are and what they are here to do. Also because they are in Cobb's mind, they know where they need to go because Cobb knows where Mal will be. Since the reason for Fischer's dying is Mal, if they find Mal, they will find Fischer and they do.

Now, they are merely in one further level of dream state below. To wake up, they need a kick in this level and one from above. Eames, from the level above, uses the defibrillator on Fischer and Ariadne pushes Fischer off the building giving him the kick in the current level (improvises). The synchronized kick brings Fischer one level up.

Fischer saw Mal shooting him and his mind accepted being shot. When he gets kicked back up, he doesn't remember the event that happened prior. Why? may be the shock of it all, waking up from a bad dream and just brushing it aside. They are all dreams at multiple levels after all. Fischer will suffer physical effects of being shot only if his mind makes that happen. In this case, his mind heads towards his dying father and focuses on that.

Saito, on the other hand, is shot by Fischer's militarized mind. Even if they went down into Fischer's mind, they would have no idea where to look for Saito because Saito was shot to protect his mind from getting attacked, Fischer has no obsession with anyone particular from his past. Besides, Fischer is being kidnapped, they would have to convince him to be the dreamer. Going down into Fischer's mind would also ruin the plan as they need to take Fischer through other dreamers deeper and deeper till the 3rd level and incept on his unsuspecting mind. The only thing they can do is hope to complete inception before Saito dies by descending further as the time dilation will give Saito more time to live. However, thanks to Mal being projected, they loose more time and Saito eventually dies and lands up on the shores of Limbo.