Under what conditions? The equipment itself is legal to be operated, so long as it meets registration, maintenance/safety, and other criteria. What is required for the driver to be operating it legally depends on the Gross Vehicle Weight Rating (or Gross Combined Weight Rating, if it's pulling a trailer), not what type of brake system it has.
is it illegal to drive with an infant in the front seat of a truck
Not if it has a Gross Vehicle Weight Rating of 26,000 lbs. or less.
That depends on which country you live in.
Yes
If the truck and trailer are not equiped with air brakes. If the truck and trailer are not comercial vehicals.
In the state of Colorado, there is no law that makes it illegal to drive a truck without doors. You would need to be sure you have a rear view mirror however.
Air brakes have absolutely NO effect on whether a vehicle requires a CDL or not.
Find someone who'll employ you once you have a CDL, and will offer their truck for the road test. It doesn't have to be a dump truck. You could rent a 33k single axle truck from Penske with an automatic and air brakes, and, according to the DOT, it would qualify you to drive a Class 8 dump truck.
You can drive barefoot if you wish, although it probably would not be a recommended practice.
The length from the middle of the steer wheels to the center point between the drive tandems (for a tandem truck) or center of the single drive axle (on a single axle truck).
Under federal law, it's illegal to service or repair brakes on an air brake vehicle without holding the appropriate ASE certification. That answer can't be given on here.
It is technically illegal to drive with your tailgate down in Tennessee. This is considered to be an unsecured load, even if there is nothing in the bed of the truck. This is only considered a violation.