Single sub-panel from 3 phase panel and the main breaker size
First things first: do you see any orange wires, or wires with orange tape on them? If so, you definitely have what is known as a "high leg" or "wild leg" delta system -- based on your voltages, I believe you have this, which was used historically to supply both 3 phase 240VAC and 1 phase 240/120VAC to mixed occupancies, as in the illustration below (courtesy Wikipedia/Gargoyle888):



In this system, the secondary center tap forms the split-phase neutral, with the A and C phases as the normal 120V "hot" legs in the derived split-phase supply, and the "high" or "wild" leg, while normally the B phase (this is from 408.3(E)(1) in the NEC, by the way) sits unused as it has 208V to the neutral on it.

Now that that's explained, to answer your questions in turn:


Typical 6/4 service entrance quadruplex uses PE (XHHW) insulation rated to 75&deg;C and is thus limited to 60A.  If you can confirm that the service entrance uses XHHW-2 (XLPE) insulation, though, you can run it up to a 90&deg;C rating, which gives you a 70A (some sources say 75A) max ampacity.  The XHHW or XHHW-2 designation is part of the markings on the insulation, by the way.
Is the ALU#4 cable type SE(R) or type USE (also called SEU) cable?  SE(R) cable can be used for feeders indoors provided that the bare conductor in the cable is used only for equipment grounding purposes, as per 338.10(B)(2), or if all wires in the cable are individually insulated, as per 338.10(B)(1).  However, USE/SEU cable cannot be used for indoor feeders as per 338.12(B)(1), as its insulation is not flame retardant.
Connect the feeder cable to the feeder breaker (either 60A or 70A) in the three-phase panel; connecting a load directly to panel busbars is simply not cool.
While your thought of making it so the subpanel main breaker trips before the feeder breaker in the main panel is appreciated, selective coordination is a much more complex piece of work than simply using a smaller subpanel main breaker than the feeder breaker.  Here's an article on the topic if you want a taste of the gory engineering details that you'll have to work out to do this.  You can use 60A breakers for both the feeder and the subpanel main, by the way; however, there are no guarantees as to which breaker trips first into a bolted fault (hard short).
You can tap the A and C legs from the existing 60A three phase breaker in the main three phase panel and use them to feed the subpanel; this is the most cost effective approach, and doesn't require any inspection of the service entrance conductors.


Finally, keep in mind that 60A is a very limited amount of current for a single dwelling unit.  It can be managed, though, if you are able to run the heavy single loads (dryer, range/stove, hot water, and HVAC) using whatever fuel gas supply is plumbed to the building instead of using electric heavy-load appliances, or if the heavy loads for that dwelling unit are run directly from the three-phase supply -- although in some high leg services, the B phase is limited to a small fraction of the total load, which can make this infeasible.

Is there a reason the utility won't simply replace the obsolete high leg delta service with either a 240/120V split phase or a 208Y/120V three phase wye service?



Addressing the conduit problem, the neutral, and the balancing issue:


I would use conduit bodies instead of elbows, unless elbows are the only thing that fits in the space.  In any case, make sure you have no more than 360 degrees of bends between your pull points!
The neutral coming from an overhead pole is on the bare wire in a triplex or quadruplex cable, just about always.
Phase balance isn't typically worried about in high-leg deltas; it's a concern in a wye system due to unbalanced currents flowing through neutrals, which need to be sized appropriately to carry it.