I'm afraid I've got the advantage on this topic as I've actually written magazine articles on several of them.
To pick up a few points :
Moscow is the most heavily used in passengers per day. London has the highest average fare per mile. New York has to be the most imnproved over the last 20 years.
Soviet systems (Moscow and St Pete are exampled above but there are others) were indeed built very deep as joint transport and nuclear bunkers. If you look closely at station entrances you can sometimes see the blast doors still there. The cars are standardised (the factory is in St Petersburg) across all Russian systems and also old eastern Europe such as Prague. The first Soviet cars, type A, now long gone, were built in the 1930s, they were a direct copy of what New York was building at the time, they got their hands on the St Louis Car (long-time subway car builders) blueprints.
Deepest station in the world is Park Pobedy in Moscow, over 300 feet (100 metres) down.
Many cities have 2 sizes of train. London has already been described, Berlin is the same, so is New York (IRT trains smaller than BMT/IND) and others.
Most subways run at about 600 volts DC by an outside third rail, the current returns to earth through the car body and tracks. London however has a fully insulated system, positive outside, negative between the tracks, still 600 volts but +400 on the outside and -200 on the centre rail. It gives an advantage in avoiding interference with the signals.
Chicago cars are the size of the old streetcars and the ones built in the 1950s were actually rebodied streetcars, then the infrastructure built around them prevented anything larger.
Britain has built subway trains for other cities over the years, like Toronto (the old red trains) and Hong Kong. In contrast the latest Jubilee Line cars in London were built in Spain.
Oldest station is Baker Street in London (Circle Line), opened in 1863, restored on its 125th anniversary to look like the old days.
Favourite systems - Paris, New York and Moscow, in no particular order.
Greatest waste of money- LA Red Line I am afraid.
Smallest "proper" subway - Glasgow.