Does a gearbox increase range?

Ummm, there are two sides to this and it depends on the application.

If you are using a dc motor without a gearbox, ie direct drive with only one ratio to the wheels then at slow speed the motor will draw more current and work less efficiently. Driving slowly in traffic like this would reduce the range a little due to the lower efficiency.
However, if you drive at consistantly high speed most of the time then the motor will mostly be running at the upper end of it's speed range and will be more efficient and so range may be improved due to the higher efficency.

If you had a gearbox giving a range of ratios then in slow speed driving a lower ratio can be selected to keep the motor at the upper spped range adding to more efficient running.
However, a multi speed gear box has a lot of moving parts adding to mecanical drag that ls less efficient then without. also there is extra weight to carry around as well and that all reduces the efficiency.


Ultimately there are gains and losses either way and it would epend on what your application is, the sorts of journies you will take and the road conditons you will be driving on.

Well, with an ICE you need to have a gearbox because the torque curve isn't very good for moving vehicles around. There is none at 0rpm and not a lot until 1000rpm, then it peaks at about 2000rpm and then falls off at 4500rpm. To make best use of that you need gears to keep the ICE spinning within that 2000-4500rpm. The closer you keep it to that 1500-2000rpm, where the torque is at its peak the more efficiently it will run.

With an electric motor the maximum (or near maximum) torque is at around 0rpm, however, the motor is at its most efficient when the torque is zero. That happens when the motor is at its highest speed when the back emf of the coils oppose the inward flowing current. The higher speed, the greater the back emf to the point where it directly opposes the current flow and the current almost stops flowing and torque falls to zero.
At this point the motor is still spinning very fast but is using almost very little current and so would appear to be very efficient.

So the closer you can get the motor to spin at its highest speed the more efficient it is. But you won't really get there as you will still need a little current flow to maintain some driving force for your EV, but it will be as efficient as it gets.

Many EV cars will keep a gearbox for lots of reasons. Reverse gear is one reason. The other reason is that it is a good way to link the motor to the wheels, especially on a front wheel drive car.
However, it is also useful to keep the motor efficient.
In town a driver may use predominantly 2nd gear right up to, say, 40mph. Then change into 3rd gear right up to 70mph. 1st, 4th and 5th may never get used and some will go as far as removing the redundent cogs.
The sort of ideal gearbox in an EV maybe a two speed and reverse in that case. I would like that as an option on my trike and so I have a two speed transfer box from a small 4x4 as an option. Low range for urban driving, high range for the motorways.


AC vs DC. That is a whole big subject.
Simply put:
AC allows higher voltages , and hence lower currents and possible saving in losses.
AC allows easier regeneration.
AC needs a more costly and complex controller possibly with more losses.
AC easier to reverse direction.

DC is easy to do.
DC components are cheaply and easily available.
DC is more difficult to do regeneration with.
DC easy to reverse direction but reversing contactors are costly (hence keeping reverse gear in a gearbox), BUT, if brush advance is used then reversing isn't so easy due to arcing.

For the novice without deep pockets it is easier to use DC as motors are found cheaply in forklift trucks and controllers are affordable.