Did either side during the Franco-Prussian war make significant use of repeat-action rifles like the US army did in the last year of the US Civil War?
Did the US make significant use of repeaters? Are repeaters a significant increase in combat power? The answer to both is not really. The number of repeaters issued (and privately purchased, the Henry was not an issue weapon, and Ordnance did not supply ammunition for it) to the Union Army was 12,472 Spencers (you can find figures of 48,000 and 90,000 on various websites, these being the number ordered during the war (the latter including the Burnside-Spencer), but Spencers were only being built at about 3-400 per month), less than 10,000 Henry's were built (at about 250 a month once they got going) by wars end, and only a portion of those (maybe 5,000) found their way into military service (the Henry was never designed as a military arm, and it was sold on the open market. It fired a pistol round...), and the last repeater, the Colt, was supplied to the Army early war (ca 4,200 of them), but was quickly withdrawn from service.
So, at best (assuming no losses etc.) by wars end maybe 15,000 repeaters are in Union use. Accounting for losses, and a fair number that were never issued (a few thousand Spencers languished in the Washington Arsenal) this figure is probably closer to 5-8,000. These are almost entirely confined to Cavalry and Mounted Brigades, and the majority of that arms weapons are breechloaders, not repeaters. Of course, they get a lot more press than their actual numbers and effect would be due.
As to combat effectiveness, over a sustained firefight a repeater of the period is actually slower firing (once reloading is taken into account) than a breechloader.
Had any European war made as much use of railroads to move and supply troops as the American Civil War?
A lot more actually. The US railroads suffered in that they were not integrated into the Telegraph system as happened in Europe, and simply operated on a timetable. By comparison, European Army commanders could control rail movements directly by Telegraph, enabling large scale flanking movements by rail for example, of the kind that never really happened in America.
If you think that there was not much trench warfare during the ACW, you've obviously never read anything about Richmond and Petersburg in 1864-65, where the largest armies on both sides were locked in a trench stalemate.
I think both of those are after Spotsylvania Court House ;-)
In fact, this kind of warfare occurred earlier (Vicksburg, even the Richmond Defences in 1862), but trench system is really a misnomer. Rather than a dug in trench system a la the western front, the field fortifications (in line with European fortifications) consisted of built up (rather than dug in) ramparts, typically with a trench dug in front of the position (as a dry moat), and a field of fire cleared out to about 100 yards, with wooden obstacles placed in the way to delay the attacker.
Did the British really fight in a loose order during the Crimean War? The few accounts I've read said that tactics had changed very little between the Napoleonic Wars and Crimea.
The open order of the Crimea was the same interval as the Napoleonic wars (the three main orders being open, close, and doubled (4 ranks, used where there was an extreme cavalry threat, such as at Waterloo)), which one they'd use depended on the threat, and the ability of the British unit in question. Close order is inherently easier to maintain than open order, which is why volunteer units tended to use it (not must the Union, Canadian Militia had extreme difficulty in fighting in open order for example, and the inexperienced battalions at New Orleans also came on in close order). By 1854, with increased firepower (particularly shell firing artillery) it was normal to fight in open order.
By your own account, the Prussians were still using very close infantry formations during the Franco-Prussian war - it seems very unlikely that they would still be doing this if other European armies had adopted loose formations 15 years earlier.
Because Prussia was highly conservative and stressed command and control over all else. Perhaps this is a natural consequence of the fairly inexperienced nature of the average Prussian conscript. Line troops were forbidden to lie down, or even kneel, in the presence of the enemy, and the French (who like the British had trained the entire infantry in light tactics, the French called them "Algerian Tactics" (from their 1830's campaigns in North Africa), the British used to call them "American Tactics" (dating back to the SYW), they entered the German lexicon in ca 1900 as "Boer Tactics" (from observations of the Great Boer War).
A normal early FPW Prussian attack consisted of the Prussians advancing in Battalion columns, with the front rank firing from the hip as they advanced. Not surprisingly, well trained French infantry with Chassepots slaughtered them.
The tactic that developed in the Union Army was a copy of Napoleonic British practice. A Regiment would normally be formed with 8 of their 10 coys in close order, with a coy (sometimes two) pushed about 150 yds forward as a skirmish screen (what we'd recognise as the light coy), and another coy in reserve about 100 yds back (what we'd recognise as the Grenadier coy, although their main use was to stop the main body straggling)