Why did the U.S. Army go to the Trapdoor instead of a better alternative? It is well known how much the Confederates hated going against the Spencer, why switch from a repeater to a single shot?

For one thing, the original Trapdoors were made from 1861 Springfield rifled muskets that had been modified to fire metallic cartridges.  They had thousands of them in stock, so it made sense (read "cheaper") to use what they had instead of starting all over.

I also heard that the Army decided against lever-action rifles like the Spencer or Henry because the lever was difficult to use when shooting prone.

The 45-70 also had a longer range and stronger punch than the 44RF in the Henry or the 56RF in the Spencer.

I'm not an expert though.  This is a combination of what I got from various books or the History Channel.  Maybe someone else knows more.

That is pretty much it.  The decision was fiscally driven, plus it was an armory design.

I.e. Springfield

OK

Also, logistics. Keeping ammo for several types of repeaters that the Army had on hand would be difficult when fighting out west where long distances were involved. It would be easier to keep standardized ammo (45-70) for one cheaply produced rifle, especially for large numbers of infantry. Non-standard ammo wouldn't be as much of a problem if you were only a few miles from a depot, as in the Civil War. Repeaters use up a lot of ammo, which sorta taxes the old ammo wagons a bit on a long campaign

I guess the ammo thing would make some sense. I heard once that Army officers usually only carried 12 rounds into battle.

One other thing I just learned about the carbine version of the Trapdoor that the cavalry used.  While it could use the 45-70 cartridge if necessary, the Army had a 45-55 loading specifically for the carbine.

I am guessing that the reason for this is that the 45-70 would create excessive recoil in the  lighter carbines.  I don't know if the cartridges were shorter, or just loaded with less powder.

This would create a little more complication in logistics, but not as much as if the other combinations of rifles and carbines had been used.

.45/55 was just loaded with less powder.

The thing is the Army was already issueing two cartridges, .45/70 for rifles and carbines, and .45/30 (.45 Schofield) for use in Colt and Smith&Wesson revolvers.  As we know today, there is no reason that the 1873 Winchester couldn't have been ordered chambered for the .45 Colt revolver cartridge, and thus been able to chamber the Government revolver ammunition.  This would have solved the problem.

It would have also helped if we had ditched the Trapdoor in 1879 and replaced it with the Remington-Lee, which we did issue on a trials basis, and was used in combat by US Infantry against the Apache and by the US Marines against Panimainians and Chinese.  Neither service decided to buy them in quantity.  Ten years later though Christopher Lee's design reappeared again- as the .303in Lee-Metford Mk1 and would remain the standard British battle rifle for another 70 years...

The decision was based on money.  Post-war, no money for the Army.  Big time RIFs.  Typical post-war cutbacks.  Use a modified standrad rifle you already have, or buy new ones?  In that economy, you go with the cheap solution.  

Also realize that there was a huge bias against a high-firepower, small caliber shorter range shoulder weapon.  Sound familiar?  The Trapdoor was a "real" rifle, in a "real" rifle cartridge.  The Henry was a "mouse-gun".  That type of bias runs deep in the military against trying new things and abandoning the old ways.  Then as it was today.  The same arguments used against the M-16 were used against the repeater: Troops won't aim, troops will use too much ammo, troops won't be able to engage targets at 2000yds, etc.  Combined with the cost of new rifles and the repeater didn't have a chance as a general issue weapon.  

In the end the Trapdoor was cheap, worked, and was fast enough for use in the west.  There were better weapons to be sure, but the outcome wouldn't be dramatically different.

Ross