It depends, there are also many power plants that use diesel engines.

Knowing about ship propulsion, compared to turbines, diesels have higher efficiency and longer times between overhaul. Turbines are lighter. Both mean that if you use turbines, you pay more for fuel and your ship must be docked more often which are a big nono in commercial shipping.

Generally only the military uses turbines in ships.

Sometimes they are installed in for example cruise ships if they need a lot of power and there's no space, / there are balance issues with diesels.

Probably on the land side if you only run it rarely, or your natural gas is extremely cheap, then a less efficient turbine is fine. Generally power plant frame turbines are more old fashioned and less efficient than aircraft turbines.

There's a difference between turbine and jet engine. Natural gas peaker plants are very expensive to operate due to fuel cost. Most big power plants do use turbines to spin the generators but the turbine is fed steam, not hot combustion gas as in a jet engine.

That's a great question. I do not know for certain, but I suspect the answer lies in reliability and much longer intervals between servicing and overhauls.

Piston aircraft engines have a recommended time between overhaul of 1200-2000 hours (with outliers above and below).

Jet aircraft engines have a recommended time between overhaul of several times that, with required inspections typically at slightly longer intervals than piston engine overhauls. (In a fixed install turbine plant, you might be able to skip the inspections as well, and just use trend monitoring data to determine when to take the turbine out of service for maintenance/overhaul. In an aircraft carrying people, the risk profile is a little different.)