The order of the perks granted by Specialist is the order they are selected in. For example, I can set it up so that Assassin is my first extra perk, Extreme Conditioning is my second, and Stalker is my third.
The only restrictions placed on what perks may be selected are the three perks you already have selected outside of Specialist, and perks that you have not yet unlocked.
Just deselect all of them, and then select them in the order you wish for them to be applied when getting kill streaks.
The more damage a gun (or class of guns) does, the more likely it is to be able to shoot through objects. That means that LMGs, assault rifles, and sniper rifles are better equipped to shoot through objects than shotguns, SMGs, machine pistols, or handguns. This doesn't apply to launchers, which inflict massive damage but are effectively useless for penetrating objects.
If you want to better shoot through objects, pick a LMG, assault rifle, or sniper rifle and use it a lot in order to increase your weapon proficiency level with that particular weapon. The more you use it, the more proficiency choices you unlock -- including one called Impact, which gives your bullets increased penetration. It's effects are similar to Hardened in Black Ops, Deep Impact in MW, and Full Metal Jacket in MW2.
As for aerial weapons, things that deal bullet damage (the helicopters) are more likely to penetrate objects than the ones that deal explosive damage (a predator missile).
Ultimately, even with with the proper gun and Impact equipped, it is going to take some trial and error to get a feel for when you can shoot through stuff, as some objects are simply easier to penetrate than others, regardless of appearance. One example that springs to mind are the concrete pillars in the center of the map Interchange; they are notoriously easy to shoot through even though they shouldn't be.
Best Answer
You can increase the performance of the weapon through weapon proficiencies (weapon perks), but there are no passive bonuses based on weapon level.