I agree with others that switchers are a better choice in terms of efficiency, but they can be somewhat complicated to deal with if you're inexperienced, and there can be lots of weird effects that aren't immediately obvious (precharge sinking, beat frequencies, etc.) that can make life difficult. Assuming you've figured out your power dissipation and know how much current each rail can deliver, if the linears will work for you, stick with them (at least for the first pass).
If you're trying to achieve a variable-amplitude square wave output on your adjustable rail, the chopping may introduce noise into the main 24V rail, which could show up on the other rails. You may want to have an LC filter between the main 24V rail and the regulator input to provide high-frequency isolation, and will probably need extra capacitance on the adjustable regulator output (bulk electrolytic as well as low-impedance ceramic) if you expect the square wave edges to be sharp.
1, 5) There are some dangers with your scheme.
Power dissipation in the linear regulators will be
\$(V_{out} - V_{in}) \cdot I_{out} \$
which is significant, especially for the lower output rails. 78xx-type regulators have built-in thermal protection around 125°C, and (without heatsinking) a junction-to-air thermal resistance of 65°C/W. Your thermal management will be challenging.
Another potential problem - if the series-pass element in any of your low-voltage regulators fails or gets bypassed (shorted), you'll present the full 24V input to the output. This could be catastrophic to low-voltage logic. You should protect your low-voltage rails with SCR crowbars that can sink enough current to put the DC/DC brick into current limit and collapse the 24V rail (they'll need big heatsinks too). Fuses are unlikely to be good protection since the 24V brick likely isn't stiff enough to generate the \$I^2 \cdot t\$ needed to blow a fuse.
2) Whatever floats your boat.
4) Meters aren't huge loads. Just use one of your rails.
3) Correct - all regulators have headroom requirements. If you want the maximum 24V out, you'll need a direct connection, and will have to rely on whatever intrinsic protections the brick will provide you.
Best Answer
Let's have a look at datasheet Figure 10 "Dropout Voltage". At 500mA it will be a bit above 1.5V.
When operating in this mode, the output pass transistor will be fully on (saturated in this case, since it is a NPN bipolar).
If this was a PNP pass device LDO, you would expect excess ground current, as the regulator attemps to saturate its output device. However, this one uses a NPN, so the excess base current will simply go into the output. No problem here.
The regulator will not regulate anything though, this means it will act either as a resistor or as a couple of diodes in series, so output voltage may vary depending on current draw. Also, output voltage will follow input voltage, so input noise will not be suppressed.
If your load works on 19V, and you have 24V input, and the load can tolerate the regulator not rejecting input noise, then you're fine. In this case it would simply act as a voltage limiter.
If you also want to filter noise, then something like a capacitance multiplier with an output voltage limit would be more suitable.
EDIT: Example
simulate this circuit – Schematic created using CircuitLab
This is a simple capacitance multiplier. It lowpass-filters the input (RC network) to filter out noise. Zener limits the voltage. I put in a CFP (double transistor) for lower output impedance, so you can say that's the "luxury" version!