Can you take the CDL road test in an automatic truck with air brakes?

A new (as of June 2014) Federal regulation requires states to put a "no manual transmission" restriction on the CDL of anyone who road tests in a truck that doesn't have a fully manual transmission.

There are two other new restrictions: if you road test in a vehicle that has "air over hydraulic" brakes (kinda like the power brake system in a car, but the booster chamber runs on air instead of vacuum because diesel engines have no usable vacuum) you will receive a "no air brakes" restriction, and if you road test in a combination vehicle that connects the trailer any other way than via fifth wheel you will receive a "no tractor trailer" restriction.

OTOH there is, so far as I know, no rule setting the minimum length of the trailer you must use in your road test.

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Yes, you can. USXpress Enterprises uses automatic trucks, and hosts their own driving school. However, I would advise against this - if you're going to be driving commercial vehicles, learning how to shift an unsynchronised transmission is going to be an important skill to know, and it's a lot different than shifting in a normal passenger vehicle. If you go to take a road test for a company which uses manual shift trucks, and you're constantly grinding gears, show yourself to be unable to downshift proficiently, can't do basic gear recovery, etc., you're not going to get the job, because no company wants to hire drivers who they know will end up destroying their transmissions.