Wow, a Crouse Hinds panel where both Eaton BR and Siemens MP/QP breakers are approved. (see the first full width paragraph halfway down). You don't see that every day.
I have the 4-prong plug/enclosure and the circuit run distance will be about 10 feet.
Well done!
Do I need a specific breaker aside from it being 50A capable?
You need to use the breaker that the dryer instructions are telling you to use. I seriously doubt it's 50 amps. But on the upside, you can run wire appropriate to the instructions and breaker, so there'll be a cost savings there.
The reason is that UL doesn't certify the dryer for circuits larger than specified. E.G. they've made sure any foreseeable failure will trip a 30A breaker, but they can't promise it'll trip a 50A.
Do I need to use a specific side of the panel?
Yes. You need to be very careful about voltage to ground. Certain 3-phase configurations have a higher voltage-to-ground on one leg, also called the "Wild Leg". An example of wild-leg is seen on the panel diagram label, to say IF you are supplying wild-leg delta to this panel, the wild leg must be on phase B. However, your panel could nonetheless be simply 120/208V "wye" arrangement with equal legs.
I notice that NONE of your phase wires are marked orange. When wild-leg is supplied, Code requires the wild-leg be orange. So I'm going to bet you don't have wild-leg. If all phases are equal to ground (probably between 110V and 130V), then you're fine and you can place the breaker between any two rows. If you mean to put it in the 2 spaces on bottom left, that appear open, that should be fine. Use a Siemens Q230 or Eaton BR230 breaker.
You may measure 120-127V from both legs to ground and 208-220v between the two legs. That is FINE. The dryer will function just fine, but with only 75-85% power, it will "duty cycle" (switch the heater on/off) at a higher percentage of the time. Where it might be "on 30 seconds, off 30 seconds" on 240V, it'll be "on 40 seconds, off 20 seconds" for you.
Now, I see a great many "Multi-Wire Branch Circuits" aka MWBC aka shared-neutral (2 hots 1 neutral) on what appear to be old-fashioned tandems with unusual handle-tie arrangements. You must be very, very careful in this panel with moving any of that stuff around. On any given MWBC (black-white-red sharing a cable), there must be 208-240V between the black and red. I personally would use black-red-blue phase tape to tag each MWBC's wire for the phase it belongs on. If you want to see what phasing is about, read this Q&A here and let me color your panel similarly to that link.

I tinted the black phase a bit purple for visibility.