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.
I'd like to start all over and make a proper working prototype on the
bench before making another board.
A better idea might be to resolve the issues you have with the TI devices/PCB. What you might find is that your assumptions about power layout on that PCB were incorrect but you'll never realize that if you move on to a different design and you'll quite possibly carry that erroneous thinking thru and find your self in exactly the same situation.
As for a proper working prototype on the bench - it's unlikely that anything other than a decently designed PCB will suffice and you will be left scratching your head once more.
I'm 56 and I use glasses and magnifying goggles to help me to solder 0603 components and SMT chips so you should consider that this sort of equipment comes with the territory of the technology.
You might find some non-SMD parts that are easier to breadboard but also you might find that these don't perform adequately if you don't use a circuit board designed to make this circuit work.
Best Answer
As AndrejaKo mentioned, you may be able to find a LM7805 (or similar) linear regulator in the 5V standby section of the PSU. Look for it near the end of the purple wire.