The display is on only when the furnace is burning gas and the blower is on. Is thisthe normal operating mode?

Yes - but it doesn't have to work that way. There are several methods you can use to have the humidifier run only when the furnace runs. Look on the back page of the installation instructions. The method I chose was to install a sail switch. This is a switch with a arm attached which has a "sail' attached to it. When the furnace blower runs, the sail catches the blower airflow and activates the switch; allowing the humidifier to operate.
You can also wire in a current sensing relay (figure 6 on the instructions) or connect the humidifier directly into the furnace motor ignition box (if so equipped. My furnace dates to the late 70's and has no place to plug in a humidifier; which is why I installed a sail switch.)
Now however I probably would go with the current sensing relay option as this looks to be the easiest route to follow. Honeywell has their 32001754-001 avaliable for less than $35 from Sears.

Hi it could be a faulty fan control it cuts in on temp rise bring the fan on / and shuts off if the temp gets too high if fan dose not cut in .

It is possible that you did not set the new blower on the proper speed. Black wire is high. Blue is medium. Red is low. If the fan is operating to slow it can run to hot and trip the high limit control. Make sure the filters are clean. When the old motor went bad, the problem might not have been discovered rite away your high limit control might need to be changed.
Also anytime you work on a furnace you can kick up some dust. Always clean the flame sensor.
Good Luck
HVAC Teacher

check the pilot gas pressure, might be too weak and blowing out when the blower starts, the clicking noise is the gas valve shutting down when it detecks no pilot flame, that park is normal.

When your unit is on normal the outside unit works and the inside only the fan runs. your heat pump does not remove any humidity when it operates so the installation cotractor did not wire it to operate then. you can change the wiring on the humidifier to operate all the time by wiring it to the fan (g) terminal then it should work just turn it all the way to the lowest humidity when you are in cooling mode.

Hi!!! You might have a bad flame sensor its a single rod located in the burners its white..If the unit runs and then shuts off and on and off.change your flame sensor..Good luckk..

sounds like wired incorrectly. always do this? not wired into blower circuit correctly

sounds like the high limits tripped

You want the humidifier to run in heat mode which is the r-w1 terminal on the stat.

the first blower you hear is the inducer motor it pushes the exhaust out. sounds like abad gas valve.do you have a standing piloit light or is it eletronic ignition. if electronic and there is no glow you have a bad ignitor