Yes, in fact it will lower your energy consumption if you have an energy efficient handler. By running it continuously it lowers the temperature and therefore the condenser will not come on as often. The condenser consumes quite a bit more energy. Handlers are designed to run continuously.
Broke
Until it runs out of gas...
If the run is less than 100' use 4 AWG copper.
Could be a bad thermostat.
It should not run continuously. Typically, the fan (sometimes called a blower) will continue to run for a minute or so after the flame is out. This does not consume much energy; it does flush the residual heated air from the ducts. Make sure the fan is set to "AUTO" on the thermostat, to ensure proper cycling.
It will run continuously if the refrigerant charge is off. The compressor, condenser fan and inside blower fan should all run at the same time. If your thermostat shows that the unit will not keep up, it will not stop. If the unit used to cycle ok, clean outside coil, change filter inside and put a thermometer on the register closest to the air handler. It should be around 52-60 degrees F.
nope
Because the air handler and ductwork are full of warm air from not running and cooling is not instantaneous upon start up. It takes 2 or 3 minutes of run time for the unit to begin cooling.
It is okay to run a dehumidifier and central air at the same time. This will help keep the costs down for the air conditioner. If the dehumidifier is running, the air temperature will cool off quicker and the air conditioner can be turned up a little higher.
You could probably run it, but you'll never START it, even if you remove the safety interlock that makes sure it wont start unless the indoor air handler portion is already running.
If it is fuel injected, the pump has to run continuously. If it has a carburetor, it is probably getting weak.
A bilge pump does not run continuously. If you have one that is running all the time it is not working properly. Running all the time will wear it out.