What do these bars below the inventory at the Deploy screen in Battlefield 3 mean

Any class and any weapon can lay down suppressive fire. The amount of suppression achieved depends on 3 things:

1. The proximity of the bullets to the person

2. The size of the bullets fired (bigger bullets, and especially frag rounds and explosives, will achieve more of a suppressive effect, so an smg will have less of a suppressive effect than a sniper rifle)

3. The amount of bullets fired (the suppression effect continues to stack until it reaches the cap, so an smg can be just as suppressive as a sniper rifle, but will require more bullets).

All of this coupled together makes the Support class generally the best class to purposely lay down suppressive fire, as using the bi-pod will make you accurate enough to get close proximity, your bullets are large enough to achieve quite a good suppression effect per bullet and you can fire much more than any other class to reach the suppression cap much more quickly.

With that said, generally, you won't purposely try to lay down suppressive fire much (unless you're in a really good squad who actually communicates and works as a team), you'll generally be taking a few shots at an enemy and a few will miss, causing suppression, and as long as they're at least a little bit suppressed, you'll get the suppression bonus if someone else kills them (they don't need to be very suppressed at all, just enough to cause the screen blur).

The best game modes to level up in would be Conquest and Rush.

The reason these two are the best are because they contain objectives that are worth a lot of points.

The fastest is probably conquest as capturing a point (including neutralizing the point first) will net you 450 points (350 for assisting a capture) and there are many many points to cap per round. In addition, if you kill anyone while inside a flag's area of influence, you'll gain an extra 30 points per kill.

Rush is the second best as you can also get 600 points per objective (100 for arming, 500 for destroying. Only 200 for defusing) played, but they're aren't as many, and it's harder to get defends etc.