The diff has the same strength regardless of gearing, but the gears themselves are stronger as the numbers go DOWN. Read your owner's manual - it should have a chapter dedicated to towing/hauling, and how to calculate the actual GCWR.

Changing the gears? Yes, and no. It's a very specialized task that requires the right experience & equipment. Read these:
http://www.fordracingparts.com/download/instructionsheets/FordInstShtRing-And-Pinion.pdf
Differential Repair & Installation Instructions

 

Going from 3.55 to 3.73 won't make much of a difference. If you're going to spend the cash on a gear swap, I'd advise going to a 4.10 ratio, especially if you plan to regularly with it. Highway mileage will drop as the highway rpms will be higher, but you'll be able to tow the maximum amount possible for your year and engine with 4.10 gears. Stepping up to a larger gear doesn't mean your towing capacity increases, as the transmission, chassis and suspension can only handle to much weight before parts begin to stress, and fail.

Lower gears allows easier starting, especially when pulling trailers. One benefit is you'll notice less gear hunting when climbing hills. Higher engine speed means less downshifting, which comes in handy when towing a load.

Don't rule out tires either. Depending on the weight of the trailer, cargo and passengers in the vehicle, you can run smaller diameter tires which will give better performance. If you run 32" tires, stock, running 30" XL rated tires (provided they fit on rim, etc) will allow the feeling of you having 3.73 gears.

 

Higher engine speed means less downshifting...

While that's technically true, it's very misleading. It's possible for the engine to be OUT of its power band on the high side, too. So if you change the axle gears & find yourself at the rev limiter often, you haven't helped performance, reliability, or economy.

...smaller diameter tires which will give better performance.

...if you consider lower speed to be "better".

...will allow the feeling of you having 3.73 gears.

Your math is a little off, and so is your theory. It won't be exactly the same - the driveshaft will be spinning 5% faster at the same actual speed, and the speedo will be +5% wrong.