Electrical – Neutral and ground connected to one bus bar in subpanel; should I change

You are probably in violation of National Electric Code, article 250.32(B)(1) states that the grounded conductor(neutral), shall not be connected to the equipment grounding conductor(ground).

The general rule of thumb is the only point the ground and neutral get bonded together is at the first point of disconnect for any incoming system, be it utility, generator, or some other source. There are a few exceptions for specific situations but none of which I feel apply.

Easiest course of actions is to kill panel, un-bond bars, drive screw through one bar into chassis and make that ground bar, use the other isolated bar for neutral. With no other knowledge of your installation I would advise you be careful of the loose neutrals while making the swap, if any are somehow being used by circuits from the other panel(it happens) they could shock you.

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.