Should the battery of a laptop be run down to its limit?

Yes... or atleast that is the correct way of maintaining the life of a battery. When you let a battery discharge completely, your resetting its chemicals back to their completely discharged state. Once you start charging again, the chemicals will change state to their "charged" state and because they have been completely discharged and charged again, you have completely restarted the chemical cycle, helping them keep their charge holding attributes.

Yes, read the user guide/manual section on battery for your brand lap top.

The battery should be fully charge the first time before running on battery power and for the first time or several time it should be run until battery reaches a almost fully discharged state.

You can run it off the power adapter 24/7 with the battery installed and fully charged as it will be running off the power adapter and wall outlet power while charging and maintaining the battery at full charge.

first element a battery desires to can charge and discharge correct, explanation why the appropriate purchase salesman informed you to allow the battery die then recharge that's it enables the battery to get use to the can charge and discharge technique, as a fashion to apply the battery to the fullest it you will desire to only would desire to do it approximately 5-10 cases that way the battery is locate to the approach, besides the undeniable fact which you do no longer would desire to save discharging the battery.