Is it possible for a turbofan jet - like on a 747 - to run without the fan, if for example you just want the thrust from the internal jet to use to heat up water, not to use as thrust?
No, the fan is directly connected to the outer compressor and turbine (N1). There is no direct connection to the inner compressor and turbine (N2), which could run independently, but there is no switch to somehow stop the N1 rotation in a 747 (and I am not sure, if it would run properly anyway).
Some turboprops (e.g. ATR-72) can run an engine without the propeller (Hotel Mode, see e.g. this question), which is connected to the engine via a gearbox. While there are also geared turbofans, you cannot completely disconnect the gearbox from the core engine.
As the other answers have said, it is however possible to build a jet engine without a fan in the first place. The Boeing 747 even has one in the tail: the Auxiliary Power Unit (APU), which will provide electricity and bleed air while the main engines are not running.
Yes, it is possible, in fact it is pretty common.
It is called a turboshaft engine or gas turbine. You can find examples of their use in:
An acronym for Auxiliary Power Units, these are probably the closest to what you have in mind, this is a jet engine, usually quite small, that is used to provide onboard power and sometimes hydraulic pressure. You can often find these located in the tail cone of modern airliners, with the exhaust at the cusp of the cone.
Most helicopters worldwide are powered by this form of jet engine, optimized for mechanical output, since helicopters generally do not benefit from exhaust jets. Otherwise it is very much a jet engine that drives a helicopter rotor (via multiple gearboxes to reduce rpm) instead of a fan.
Just like the helicopter version, although with different optimizations. You can find these in the M1 Abrams and the T-80, among others.
Electrical power plants
The generation process at most modern gas-fired electrical plants involves a gas turbine that works like an upscaled turboshaft, although usually in combination with other systems.
Yes. A turbofan without the fan would be called a turboshaft engine.