Is some asymmetry in a breaker box normal? What about neutral to ground having voltage?
Neutral-ground bonding

The panels are wired correctly. There must only be one connection between neutral and ground in the main service panel. 

If a subpanel has a bond, then it puts the neutral in parallel with the ground wire back to the main panel -- and this means current from normal loads (that normally the neutral would carry) will travel on ground paths back to the main panel. This includes bare ground wires, conduits, and metal housings. If someone were to touch anything grounded, they could be electrocuted. 

This can also mean that the magnetic fields created in the hot and neutral don't cancel each other out, which can generate a lot of interference. 

In the worst case, bad connections or other faults can cause a voltage difference between ground and neutral, causing a lot of very difficult to diagnose problems, and making it even more likely to be electrocuted by touching something grounded (anywhere in the house, from any panel). 

In short: There must be only one bond between ground and neutral, and it goes in the main panel.

Differences in voltage

The differences in voltage between the two busses is fairly normal. It is likely caused by the transformer on the pole, or by a load in your house that is causing a voltage drop. For the latter you can try turning off circuits to see if it goes away, but for the former, you're pretty much SOL. That said, it's not a problem. Mains in North America is supposed to be 120V +/-5% (so 114 to 126V is acceptable). 

The voltage from neutral-ground is also normal. Here is a decent (but technical) article on neutral-ground voltage. In short, it's caused by the fact that over any length of wire, you have some drop in voltage. Since no current flows in the ground wire, there is no drop there and thus you have a difference in voltage. Another source is induced current, which can be caused by wires going through magnetic fields, such as near motors. 

In short, it doesn't sound like you have any problems.