There is a general principle that you should apply.
When you run into a "funky" circuit you must assume that proper circuit principles may have been (were probably) violated. There are a couple of things you should do after you make that discovery.
1. Since it is a "funky" circuit it is probably non-standard and nobody here is going to be able to tell you how it is really connected.
2. Given the Situation #1, your only solution is to trace it out or figure it out, and draw it on paper, until you KNOW that you understand it. You MUST verify your paper diagram by measurements and by verifying the sources of the power and neutral wires. You must understand how it works as installed. If you must disassemble a circuit to check it out, you MUST label all wires so you can put it back together.
If you aren't prepared to do #1 and #2, then you shouldn't mess with the circuit.
If you want to just disconnect the black and white wires of an old light fixture and connect the black and white wires of a new fixture, and leave any old errors in place, then you can do that. However, you must realize that if there is a problem and the customer or the insurance company or the fire marshall bring in a licensed electrician they are going to want to know who last worked on the circuit.
..takes down notes on that one to remember... brilliant answer Bob!!