Using a 12V battery while simultaneously charging via solar panel

1. It's common for a voltage source to drop in voltage when current is drawn. This is down to the source resistance inside the voltage source. When you measure the output voltage of the panel without it connected, then you a measuring the voltage of the panel without any current passing through its internal source resistance. When current flows, it passes through that source resitance and there is a voltage drop across it, so the output voltage now drops.

How much the voltage drops when under load depends on how much current you are trying to draw and the internal source resistance of the panel.

a) It could be natural and a function of the solar panel itself. b) You could be drawing too much current from the panel and the voltage drops greatly.

2. Battery discharging: it could be that the battery is pushing power back into the panel. You can stop this by putting in series a suitable diode. You'd have to measure voltages and currents to work out what it is going on, measure the battery voltage with the panel disconnected at this point. An ammeter in series will tell you which way the current is flowing and hence which voltage is greater than the other and this will help you work out what is going on.

3. Answer: Yes.

4. I can't answer, I don't actually know what's defined by 'bulk charging'.

Should work. Cheap charge controllers are a bang bang MOSFET switch, so either the battery is straight on the solar cell, or not. MPPT controllers have a switch mode buck converter; two of them on one panel will get very confused.

Once the battery is full, the charge controller switch will be mostly off, so the other charge controller can use the solar cell.

The danger might be that if both batteries are low, both controllers will come on together, shorting two batteries together and damaging the controllers. You could check for diodes or add them. They must go before the controllers, not between the controller and the battery, as it needs to sense the battery voltage.

The open questions are 1) are there already diodes in the charge controller, and 2) what does the controller do when the solar voltage collapses to ~14V (when the other one shorts the panel to the battery). It might panic and not charge.
It's worth asking the supplier about both of these, I'm sure they've had these questions before