Does a whole house humidifier go on the intake or outlet of the furnace

It does not matter if it's on the skinny or wide side, as long as the surface the unit mounts on is 10" wide. There may be issues if the duct were say, 3x10, but any duct large enough for the whole house will be fine with the unit installed on any surface at least 10" wide.

You've asked several questions here.

Checking if it's 24V AC

First of all, by far the vast majority of furnaces use 24VAC for control wiring. However, it's always better to be sure rather than make assumptions, so there are several ways to do this:

• Look at the spec sheet or manual for your model
• Look for labels on the control board, probably where the transformer lines come in
• Look for labels on the transformer itself
• Use a multimeter to measure between C and R terminals

Update:

Since you mentioned you have a 9GMXT furnace, it has a 24VAC dedicated humidifier terminal. The manual states:

The 24 VAC HUM terminal is energized when the low pressure closes during a call for heat. [..] Connect a 24 VAC humidifier to the 24 VAC HUM terminal and C screw terminal strip on the control board thermostat strip.

It's rated at 0.5 Amps max. The Desert Springs motor is 24VAC 4W (I had to go check on mine) which is 0.17 amps, so it's perfectly fine.

Connecting

In general, by connecting between R and C, the thermostat motor will only have the ability to come on when the furnace is actively heating (it won't come on if the fan is set to 'On' mode, or if the air conditioner is on). Since you have a Humidifier terminal, instead of R you'd connect to that.

You should also connect the humidistat that comes with it in-line with this wire (doesn't matter wire is which) as per the instructions.

Water line

One more note: don't use the saddle valve that came with it (or if you do: install a proper in-line ball valve and never turn the saddle valve's "valve" again). Saddle valves are notoriously bad and will leak after you use them a few times. Better to just buy a valve with a 1/4" compression fitting and use that.

I installed the same humidifier (DS32000) a couple years ago, including the auto-flush valve. It has been working great and I'm very happy with it. It keeps the humidity level very consistent -- though I kind of wish I had an electronic control for it that would adjust based on outside humidity/temperature too (EDIT: I now have this, see below): as it has gotten really cold out in the past few weeks, I've had to turn it down to avoid condensation on the windows.

Each year I use some vinegar to clean out the valve, basin, and drum (tip: I take the curved plastic lid off, turn it upside down, pour in a mixture of half vinegar, half water, then place the drum in it and rotate it every 10 minutes or so until the whole thing is done). Better to do this at the end of the heating season.

EDIT: This year, I installed an Ecobee thermostat and ran an extra wire so it controls the humidifier. It has a setting to automatically regulate humidity based on outside weather (which it gets via internet, not a real sensor, unfortunately), and it has actually worked really well. No more condensation, but it kept the humidity as high as it could. I adjusted it a bit when I first installed it but haven't touched it in months.