Can say I read it completely
Logic analyzers generally allow any signal to trigger reading in the states of all its inputs. This can be rising, or falling edge, and can be multiple input conditions.
The limit of how much it can 'capture' is down to its memory, so the faster its input capture, the shorter the period it can capture. Because of this, often you want to run it at the slowest speed possible to capture the events you're interested in. For example, serial, at 9600, then you could get away with 100us, but probably want 20-50us. But if you suspect glitches that can be two rising edges that are meant to be synchronous, you can ramp up the speed and capture events in the range of nano seconds (depending on the max capture rate).
You mention several signals, most seems slow, so you might not need to capture in the MHz range, but if there is say 5 seconds between two events, then if you capture 8 channels at 1us.. in 5 seconds thats 5kbyte's. Often analyzers will have much greater memory than this but you see my point. Most modern small analyzers simply capture and buffer the input states and send them to a PC - used as a nice GUI. The PC can of course store this, and if the datarate between the analyzer and PC is faster than your capture rate, then the PC will capture *all* the data for however long you run it for.
In terms of voltages, whilst there are proper methods for scaling down voltages, not logic analyzers have fairly high input impedance, so you could probably get away with a simple resistor voltage divider on the inputs that are greater than 5V, or just a series resistor with diodes to clamp the input to 5V +/- 0.6V. This can slow down edges but as you're not looking at anything particularly fast it shouldn't matter too much. I imagine most analyzers will have input protection anyway, so you could get away with just series resistors - but this is no guarantee. Best bet would be a simple NPN transistor with a pull up. This will invert the signals of interest, but does the job (and makes it easier to read RS232 bytes as they are inverted anyway).
Depending on how electronics savvy you are, it might be a better idea to use a microcontroller to capture the serial, the frequency and possibly duty cycle of the encoder, and log other trigger events. Each time it captures any data (for the serial) and adds a time stamp. Stream this to the PC so it can log all the events in a CSV/text file. This doesn't actually solve any problems you may have, and would spit out very large files (read: lots of scrolling to find anything) but at least it will all be logged. For the msot par though, I'm sure a logic analyzer would be the first port of call and I'm sure a Saleae is more than capable of continuously capturing events and storing to a file for later analysis.
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