That's why road diets and reducing road capacity is also important. Like Amsterdam, Copenhagen, Madrid or Vienna, the space dedicated to cars also have to be reduced in order to incite people to use something else than a car.

Standard guage rails are 1.435m wide, plus surrounding space because trains are wider than rails plus a second track.

Not a problem. Stations where a lot of people board and unboard at the same time have sometimes platform where one side is for boarding, and the other for unboarding. Plus, trains can have more doors per car.

Usually these systems rely on people getting on/off at different stops, rather than one stop seeing full volume. If it's one stop, chances are it'll look like a terminus station and you'll need several platforms and possibly dual-side boarding to each train. It'll be quite a bit wider than tracks with no station, or a minimalist station.

This is pretty common at major sports arenas.

The same of course applies to other transit options: high-capacity bus stops take up space, and motorway interchanges and especially carparks also take up a lot of space.

Yes, these kinds of transfer numbers are easily possible (even though other posters have said you don't actually have 1,000 getting on or off at one stop). As an example, consider the subway of Toronto, Canada on its busiest line, Line 1. A subway train is 138m (450 ft) long with 6 cars (though there is no internal barrier between cars) and a capacity of about 1,500 people. Each car has 4 door sets per side, and these door sets are about 1.5m (5 ft) wide. People can easily fit through them in pairs, so moving 4 or more people per door set when in a rush is very doable. With 24 door sets (only one side opens at a station), that's 96 people entering or exiting per second, so 10 or 11 seconds for 1,000 people. If you think 4 per second per door is too optimistic, then it's 1,000 people in 20 seconds.