My wife recently purchased an '07 F-150 Harley-Davidson, and we were curious about the benefits of an auxiliary transmission. She's toying with the idea of reducing the wheel size (currently stock 22s).
Are the auxiliary transmissions effective at increasing mileage without sacrificing merging/passing acceleration (or maybe even improving it)?
You might want to check with the folks at U.S. Gear, makers of the popular Dual Range auxiliary transmission. It's basically an electronically controlled 2-speed gearbox that bolts onto the rear of an automatic or manual transmission, and provides a separate range of gearing which applies to all the forward gears of your OEM transmission -- kind of like swapping the gears in your rear axle with the flip of a switch.
The Dual Range auxiliary transmission comes in two different styles: Underdrive (separate range of low-end gears) and the Overdrive unit for additional high range gearing.
A 4-speed transmission with the Overdrive unit installed will make available a normal 1st gear (plus 1st gear Overdrive), 2nd gear (plus 2nd gear overdrive) and so on. U.S. Gear suggests the best way to go with a pickup truck is the Underdrive unit to boost towing power. The Overdrive unit enables higher gearing than factory overdrive which can be hard on the drivetrain.
U.S. Gear also states that either unit will enhance both power and fuel economy. I haven't seen statistics to support that, but you could assume more power with underdrive and better mpg with overdrive.
If you're doing long-stretch towing on high pitched inclines and declines it may be an advantage having all those extra gears to play with. Otherwise, it might not warrant the cost (over $2,000 not including installation).
Not sure where your wife is going with downsizing the wheels. The "wheel" size is irrelevant to a final drive ratio. Tire diameter is what makes the difference, and shorter tires will increase your truck's jump from a stop while decreasing highway fuel economy.