The biggest differences between road and tri-specific shoes are obvious ones being the retention systems used by both of them and of course the stability provided by the road shoe design. The road shoe retention system is built for stronger foot retention and in turn better stability on the foot thanks to more straps and usually a stiffer heel cup and stiffer material construction. The road shoe will typically a little warmer also overall and perhaps will retain water a little longer over the open construction of a triathlon shoe. The triathlon shoe will also have a softer heel cup arguably allows for better entry into the shoe but this is not always the case. The construction of most triathlon shoes is also significantly open to improving ventilation.
The better the shoe can control the foot in the cycling shoe, the more effective the force to the pedal can potentially be. For starters, stability and rigidity in material choices allows for more effective force application. What could this mean? It means the foot has better potential to perform work. Is it always so? No. You need to also start by building a foundation under the foot which means always starting with a properly built insole. Second, the cleat setup and correct rotation is also important in helping to maintain pedaling stability and prevent overuse injury.
I think most of the reason why you see many pros using road shoes in competitions is for, if nothing else, more performance from their equipment and increase foot retention in the shoe even if they sacrifice a little with transition time on and off the bike.
You do tend to see people also wearing mountain bike shoes for triathlon. This usually happens with beginners new to the sport and or athletes not confident putting their shoes on while already on the bike. Usually, people wearing mtb shoes will actually put them on in the transition zone and then run in them to the start of the bike instead of running barefoot with the shoes already on. The downside of wearing an mtb shoe is probably the biggest factor for not wearing them in competition is the weight penalty. For the most part, you can easily give up a 100 grams per shoe very quickly from a road shoe to a mountain bike shoe which over a long distance event can cost a significant amount of energy usually needed for a run.
Actually, both road and tri shoes usually are produced out the same sole mold for sake of production cost. A slightly elevated toe in the sole of the shoe helps in activating the windlass mechanism in the foot to aid in the propulsion phase of the pedal stroke. There is more to it than simply this but in general, yes, it can help with performance.
Are carbon soles always a better choice than synthetic soles? Why (not)? Carbon soles or better worded, stiff soles can be very good for performance. Why carbon is used in the construction of the shoe is simply strength to weight can be very good. The stiffness of the shoe sole itself especially in the length provides better performance mechanically as the energy is better transferred to the pedal. More complex sole construction and more premium carbon usage also helps with torsional deflection in the sole and again improves performance and minimizes losses in torque at the pedal. Perhaps in triathlon where the overall torque profile is smoother and more even at lower peak loads, this is impacting performance less but it still has the potential to affect the performance.
An argument not to use carbon and to look to shoes utilizing nylon composites can be for extra durability to the sole if you had the tendency to walk more in the shoes but this applies to not only triathletes but cyclists in general.
This could be explained perhaps a number of different ways but perhaps a few of the most common ways is too much movement in the shoes if the shoe is too big and foot is too mobile or too compressive with a shoe too small compressing the toes. The most common of problems is Morton’s neuroma which is a nerve compression condition usually radiating from between the 3rd and 4th metatarsals. A proper functioning insole of the right material density and properly supporting the heel, arch, and forefoot usually helps with immediate effect. Material choice providing good dampening properties while providing rigid support work best and can be highly adaptable in the shoe. The interface between the shoe sole and the foot is critical. Offloading the pressure area can also be hugely beneficial and getting the cleat mounting point away from the metatarsals can aid in reducing recurrence and inflammation in these problems.
This follows from the previous question but very simply, a shoe that fits well with a stiff sole will provide the athlete with the best possible out come. A mechanism to hold the shoe level on the bike by usage of a small strap system adds convenience to competition usage and with wider fitting velcro closures will provide the foot with the best retention while allowing best ease of use.
Sure, traditionally bike shops paid less attention to having a good selection of sizing inventory but this has greatly improved over the last few years in general. A good shop will be able to provide you with 2-4 choices in models/widths/sizes to give you the best option. If you use an insole, bring it with you or ask the shop if they have options to test different types of off the shelf insoles. An insole is very critical to good fit so make sure the shop can help you out. Small pressure points in the shoe can be adjusted but in general, the foot should have a good overall even pressure and enough adjustment range in the straps to properly secure the foot.