Is it dangerous to step down from such a high voltage to reach 5 V DC?
No, this is very common. All "wall-wart" adapters do this. Use a properly certified product to do this.
Are there potential stability issues?
No, but you'll have slight imbalance between usage of phases, this won't cause any issues. If your application requires a 3-phase supply then that will certainly dominate (I assume your 24V DC and 5V DC parts draw relatively little current)
As an example, the house in which I live is in a country where almost all houses have a single phase 240 V AC supply. This house had a three-phase supply fitted as it was intended to install an electric home heating system that needed a three-phase supply. However one of those phases would additionally supply all the other household appliances, including 32 A circuits for electric ovens, showers etc. In fact I have gas powered heating installed and only use one of those three electrical phases, the other two have 0 A drawn from them. That's a fairly extreme imbalance. I encounter no stability issues.
Does it not make sense to convert between three-phase and single-phase power?
If you have a device that works from a three-phase supply it is entirely appropriate to use a three-phase supply.
Could a power supply simply be connected directly to the 240 V AC source?
Yes. This is normal. By directly I assume you mean through an appropriate distribution panel with RCD/GFCI protection and permanently wired to a suitable outlet conforming to all local regulations.
Or possibly some sort of custom step-down transformer?
That's just one particular type of power supply. So yes.
Consider a pre-assembled multiple output DC-DC converter module. I'm guessing you'd be looking for one that outputs +/-15V from 9-18V in? You can probably even find one that has a third output with 3.3V. They're usually not overly cheap, but they do simplify the design process, they at least make claims to meet a particular noise spec, and you can choose one that you're able to hand solder easily - either through-hole or leaded SMT. They're readily available from the big vendors (Digikey claim over 200,000 DC-DC converter modules on their books).
For example, the LT3467 for the +/-15V part, and an LTC1174-3.3 for 3.3V can be used (or LTC1164HV-3.3 for a little bit more flexibility in input voltage), according to the schematics suggested in their datasheets.
Best Answer
You need a 12V to 5V DC-DC step-down converter (also look for: switching regulators). You can find plenty of them in the usual electronics parts vendors.
You could also consider a linear voltage regulator, but that would mean a maximum power dissipation in it of about \$ P_d = (12-5) \cdot 1.2 = 8.4W \$, which will (almost surely) either fry it or drive it into thermal shutdown. Also, it's very inefficient power-wise. Not recommended at all.