If this is the only panel box, or the main panel box, then the neutral and ground wires may terminate at the same set of bus bars. This is where the neutral and ground wires are bonded together. What we might be seeing is a dual bus bar setup that is stacked and offset like stadium seating. They are likely permanently bonded together. There is no stipulation in the order of termination so long as only one wire is secured per terminal (unless the manufacturer specifies otherwise).

The ground wires outside of the box are improperly terminated. They should run into the box and to those bus bars. Look at the corrosion and dirt on that twisted mess. It most likely is not able to conduct fault current properly to trip a breaker. That's a fire and electrocution hazard right there.

Splices in the panel box are allowed so long as they are properly spliced and the total wire volume does not fill the panel to the point where you need to squeeze the cover on or tamp the wires so the cover fits.

Multiple cables through a single clamp is also a violation as the clamps are only rated to secure a single cable. The only way multiple wires can terminate to the same knockout is if the clamp is rated to do so. Standard clamps in the picture are rated for a single cable only. Multiple wires in a single clamp can slide against each other allowing them to pull out. So the dunderhead electrician tightens the clamp even more possibly damaging the cable in the process.

This is an older panel box which was built in a time where the NEC didn't stipulate more room for a panel box. Modern boxes have much more room. Tandem breakers or skinnies are allowed only if the number of circuits does not exceed the panels rating. E.g. a panel rated for 20 circuits which has 20 spaces (a 20/20 panel) can only have 20 circuits regardless of the type of breaker used. You could have 5 tandems and 10 singles for a total of 20 circuits. It can not be filled with 20 tandem breakers for 40 circuits or any combination which exceeds 20 circuits UNLESS the panel has a higher circuit to space rating like 20/40. This change was done in 2014 to alleviate overcrowded panels.

Replace this panel before it burns your house down!

You need a new panel not a new neutral bar, as this panel is trying to cook you, your kids, and your house as I speak. The FPE Stab-Loks are rife with trouble, and as you have seen, will attempt to burn your house down instead of doing their job. In particular, at least one of your breakers is out of calibration i.e. not tripping on a low-level overload that should have eventually tripped it, and the excess current is slow-cooking your neutral bar, and who-knows-what-else in the house. Also, that slow-cooked neutral bar isn't making good contact with at least one of the neutral wires, which is what is causing the voltage imbalance in your house, most likely.

Until then, keep your smoke detectors working and your fire escape plans practiced, as it's not a matter of if, but when this panel will ignite again...