Let me start by saying that the question of three-phase power in a residence is something which I've given a disproportionate amount of time and effort thinking about, and it's basically a dream of mine to one day do exactly what you're doing: refitting a home so it is unchained from the limits of standard 120/240 North American split-phase power. I will say my dreams go ~~majorly~~ slightly off the rails, as you'll see in a bit, but I can offer some points of consideration for your questions.
To make sure we're all on the same page, let me lay some redundant background. Single-family homes in North America are provisioned with 3-wire "split phase" service, consisting of two live conductors ("ungrounded" in NEC speak) and a neutral. At the pole transformer, the neutral is tied to a ground rod, and at the service disconnect on the house, the neutral wire is tied to the house's ground rod(s). The live conductors provide 120 volts w.r.t. ground and neutral, and 240 volts between the two live conductors. This is "split phase" because it's a single-phase service that taps the neutral in the middle of the transformer, which "splits" the phase in twain. A pure single-phase service would only need two wires (live and neutral), but there would be no choice of voltage; split phase allows American homes to supply appliances for single-phase 120v or single-phase 240v.
Used predominantly in light commercial buildings, 120/208v 4-wire service is a 3-phase wye with three live conductors at the ends of the "Y" and a neutral wire at the center. As most services in North American are grounded somewhere (eg TN-S-C), the neutral is tied to a ground rod when entering the building. The line-to-neutral voltage is 120v and between any two live wires is 208v, due to the relationship of the sqrt(3) and other trigonometric thingies. This way, the building can supply single-phase 120v devices in three different combos, or single-phase 208v devices in three different combos, or three-phase 208v devices. For the most part, appliances that accept 208v also accept 240v and vice-versa, but best to explicitly check.
Historically, the same benefits of supplying multiple voltages and phases to the customer was also achieved using 120/240 "high-leg" delta 4-wire 3-phase service. This also consists of four wires, but arranged as a 3-phase delta (ie triangle), with three live wires at each corner and the neutral wire situated in-between the corners labeled A and C. If this neutral seems oddly similar to split-phase's neutral, that's because a high-leg delta is the logical 3-phase version of 120/240 split-phase. The neutral is tied to a ground rod upon entering the building. Between any two corners is a single-phase 240v supply. Between either A-to-neutral or C-to-neutral is single-phase 120v. But between B and neutral is single-phase 208v. That's a bonus voltage! But it's also a curse, because sometimes people forget about this high-leg and mistakenly feed 208v to some unsuspecting 120v appliances. Big oof.
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As for the relative merits of 120/208 wye versus 120/240 high-leg delta, they both can supply 3-phase appliances, as well as be separately used for supplying single-phase appliances. The obvious difference is that 120/208 wye maxes out at 208v in either 3-phase or single-phase, whereas 120/240 high-leg delta tops out at 240v in 3-phase or single-phase. Higher voltage means the same power can be sent with less current or less copper. J1772 or J3400 electric car chargers all tolerate up to 240v single-phase, so supplying only 208v single-phase means leaving some charging capacity on the table. Likewise, a lower voltage means less resistive heat, so a domestic clothes dryer that specifies 240v single-phase will dry a bit slower when supplied with 208v single-phase, due to Ohm's Law.
As mentioned earlier, 120/240 high-leg delta has that bonus single-phase 208v, but this may not be very advantageous then, considering all of the drawbacks about 208v I just described. 120/240 high-leg delta already provides three different ways to make 240v single-phase, so there isn't really a compelling reason to intentionally use the single-phase 208v "bonus" that it provides. If anything, it's just a nuisance and footgun to watch out for. But this very much turns on what sort of loads you'll have in your home.
The prevalence of 120/240 split-phase in American homes is in-part due to its bias towards serving lots of small 120v single-phase appliances and then a handful of larger appliances that need 240v single-phase. This works out great, until you want to draw more than 1500 W from a standard NEMA 5 receptacle.
For light commercial buildings, they're assumed to have a lot more 120v single-appliances but rather than TVs and toasters, it used to be a lot of lighting. As in, ceiling lamps. So to provide power to these businesses with less resistive losses for the utility (not discussed here), as well as balancing the utility's 3-phase grid, these customers get 120/208 wye service. The idea is that every third equal-length string of ceiling lamps is between different phases. So if they're 120v single-phase lamps, the wiring would be phase A to neutral, phase B to neutral, or phase C to neutral. But if they're 208v single-phase lamps, then wiring would be phases A to B, B to C, or C to A. In either case, the service is perfectly balanced and the resistive losses are reduced for the utility by supplying the customer with 3-phase. The customer can also supply their HVAC or elevators with 3-phase 208v, which generally works fine.
The same perfectly-balanced scenario is not possible for 120/240 high-leg delta if there are substantial 120v single-phase loads. This is because -- from the diagram above -- the 120v single-phase only exists between neutral and either phase A or phase C. Balancing every other lamp onto the available 120v single-phase wiring combinations always results in underloading phase B. The utility won't be happy, and because ungrounded conductor wire sizes must be identically-sized for a 3-phase high-leg delta, this means the copper/aluminum for phase B is being wasted. This is partly why 120/240 high-leg delta is somewhat rare nowadays. However, 240v single-phase lamps can be balanced with a high-leg delta.
To be more clear, the benefit of 120/208 wye is the ability to supply lots of 120v single-phase appliances without major balancing issues. The drawback is that 208 < 240. The benefit of 120/240 high-leg delta is supplying lots of 240v single-phase or three-phase appliances, but has a lower capacity for 120v single-phase appliances and a confusing 208v.
With all that said, we can now answer your questions 1 and 2 simultaneously. Since you expect to have numerous 120v loads, but want the flexibility to also feed 240v single-phase, 120/208 wye is looking pretty good. A three-phase panel would have it so that every breaker slot is fed from a different phase, so a single-phase circuit would have one hot wire from the breaker, and another wire to the common neutral bar. This setup is easy to work with and has no chance of 208v appearing w.r.t the neutral bar. You would have to contend with an electric car that charges slight slower, and a dryer that runs slightly longer.
For question 3, I'm of the opinion that balancing phases in a residence -- even one with three-phase -- should stop when it becomes more granular than individual breakers. Obviously, three-phase loads are self-balancing. But for single-phase 120v and single-phase 240v circuits, it's a "best effort" where your electric car charger, dryer, and welder are placed on different phases, and your bedrooms on different phases. Anything more optimized than that is excessive and is barely noticeable by the utility anyway. They're probably more concerned about your power factor than the phase balance anyway.