Need help dialing in my Hampton i300

Tha Hampton i300 is basically the same as my Regency I3100L with the exception of cast iron parts and a slightly smaller firebox. The ash on top of coals is normal. If I have a lot of coals I'll rake the ash off, distribute the coals around in the firebox, open the air all the way and let them burn down to about 1" before reloading with wood. I always pull the coals forward when reloading which seems to give a better burndown of the coals. In regard to getting the most heat moved around with the fan on high, my opinion is that it depends on the layout of your house and numerous other physical factors. Naturally the high fan setting will blow hot air out away from the appliance, but that may not necessarily equate to "most efficient heat distribution" throughout. Many people, including myself have discovered that a more even heating of all space can be achieved with the lower fan setting on the appliance and room-to-room distribution assisted with ceiling fan(s).
If your fan is turning itself off when you reload the firebox (with the air wide open) my guess is the box is cooler than the preset sensor setting for the fan and most heat is going out the flue. Once you damper down to 1/2 then the heat is staying in the box and the sensor trips the fan on (simplistically speaking).
I rarely have the fan on high and never for an overnight burn. Keep in mind, the more air movement over the firebox, the more heat transfer occurs and thus the box goes cool more quickly. That is why if there is ever an "over fire" situation the damper needs to be closed and the fan set to "high"; to help cool the box. These are just my findings and not to be cast in stone.

I have this stove. I leave the fan on high all the time to pump the heat out of it most efficiently (yes, it cools down the stove, by putting that heat into the room... sort of the point).

My heat switch is not as sensitive to the fan being on, etc. Make sure that your blower piece is mounted to the little clip brackets on the stove, and not sitting on the floor unsuspended. I often do use the "on" switch (bottom switch rocked forward, rather than set to auto) before it comes up to temp.

Mine likes to be loaded to the gills. Save smaller splits to fill in the top. That very limited space up top seems to promote better secondary combustion in a big way. With dry wood, I close the damper all the way down (do it in stages so that you know it will maintain secondary combustion).

I have a 1600 sq ft log home (some reasonable drafty spots), and this stove does a good job heating it.

Yep - with less than 8' ceilings and heads more than 6' above ground level, you have a totally different situation than I do. The ceiling in my den with the fireplace is 14' vaulted and all other ceilings are 8' high. These types of differences are why I just cringe when people say they "were told" or "read" that xxx stove/insert will heat yyy sq ft house. There are so many variables that people need to consider or they may be disappointed.