Is it normal for the fan on a heat pump/air conditioner...?

Normally the outside fan will quit when the t-stat is satisfied. Also your unit should start using the heat pump before the auxiliary heat kicks in. The aux. heat is electric heat strips in the furnace. My advice is to get the installer to check out the unit asap.

I would recall the installers. We can't test it, pressures, temps, amps, etc., to be of any real help, only guess; and you'd have to call someone anyway.

The starting sequence is out of order with the aux. heat coming on first and the outdoor unit fan running and obviously the system should stop when the t-stat is satisfied. That's all I can say for sure about your system operation from here.

they got it wired backward should be h/p first then aux heat its two stage heat heat strips should kick in second the out side should kick off then a/h do you have digital t-stat? sounds like its set up wrong thats how mine works i do this for a livin hope this helps gl

You have problems with the system, get the installing company back out and get it corrected.

My aircons fan does stop when the correct temperature is reached.