Being below weight capacity proportionally increases power recharge speed. Being above it decreases it. Carrying less guns reduces your weight (thereby increasing recharge speed). It does not affect the amount of ammo individual weapons hold (mods do).

The wiki weapons entry doesn't currently have individual weights for weapons, but likely that information will be added soon. In the meantime on the weapon equip screen as you add/remove weapons it will give you a percentage penalty/bonus to recharge rate. This can give you a base idea, but ultimately if you use a lot of powers you'll want to equip as few weapons as possible and put some points into increased carrying capacity.

There are no weapon unlocks for Mass Effect 3, that was misinformation. All you can earn for Mass Effect 3 are War Assets for completing Infiltrator and points toward your Galactic Readiness rating by uploading Cerberus intel.

Source: Video interview with Mass Effect Infiltrator design director Jarrad Trudgen. Relevant information at 3:04.