I've answered this quite in depth in the past in the linked below post; I cover the American Civil War, the Austro-Prussian War, and the Franco-Prussian Wars in relative depth:

• We often hear about how war changed in the 20th century due to machine guns and tanks, but how was warfare revolutionized in the 19th century through technological advances? Were there any precursors to the 20th century's changes?

• How did the Napoleonic style of war (formations, cavalry charges, volley fire) die out?

TL;DR:

• It was neither 'line tactics' nor 'no mans land'.

A common misconception, a very common one at that perpetrated by certain recently super popular podcasters (be it what he said or not; it's what people take from his conjecture), is that "Napoleonic Warfare" extended to 1914. Napoleonic Warfare stopped existing, fullstop, completely and entirely, after 1815. Linear tactics as a whole, as we know them, also stopped existing and would experience its complete death in 1866 in the Austro-Prussian War. In terms of the 'wars of german unification', yes that is a real term, "line tactics" were used once and only once by the Austrians and quite disastrously at that.

• You must separate the American experience and the European one completely

They were not at this time a single "Western" entity in the military sense; their doctrines and histories were not shared. I explain this in depth in the first linked post and I won't go into it much further for that reason. The Americans did legitimately attempt to use Napoleonic tactics but failed in numerous key areas; it was disastrous and they got a crash course on why Europe stopped using them in the general by that point.

Longer TL;DR:

In Prussia's wars during this time would there still be elements of Line Infantry Doctrine? What would the formations look like?

"Depends" but infantry chains and skirmish lines ruled the day in the latter half of the 19th century. Infantry chains meaning that the battalion ceased to be the lowest tactical level; company and even platoon officers could break off and act independently given the necessity thus allowing unprecedented flexibility. Skirmishing lines were just what they sound like; they were skirmishing lines. Men would have, approximately, 3 meters on either side of them about 65-70 men wide and two ranks deep with 7-8 meters of distance between the ranks. Behind them would be the rest of the battalion in column formation who would send up 'waves' of ~70 men (a platoon) to reinforce the line as needed.

This is how, very roughly, how infantry formations looked like. The Austrians were an exception in 1866 but again I explain that in depth in my first link so I'd recommend you look there. The TL;DR is that the French used close order hyper offensive techniques to charge the Austrians in the Italian Unification Wars in short time thus making the long sighted Austrian rifles, literally, shoot over the French soldiers heads due to the quickness of the maneuver. The Austrians attempted to replicate this against the Prussians but found bolt action, breech loaded cartridge based rifles to be a bit more imposing than the muzzle loaded rifled muskets the French faced up against when that doctrine worked so well.

The Crimean War was not 'line infantry' in the traditional sense; in fact it was a war defined by earthworks, trenches, and siege warfare (Sevastapol especially). Yes men still, infamously at Balaclava, lined up in something representing 'a line' and, often, did stand relatively closely. However it would be wrong to call the tactics napoleonic or even 'line tactics' because they did have distinct differences; they were far more flexible and becoming increasingly more thin with emphasis on a constant stream of reinforcements in waves with a thin but flexible front to allow dynamic reaction. The Crimean War would be the first 'real' war since the Napoleonic Wars in terms of major powers getting into it and would be the crash course that knocked any vestiges of the past out of them. However just because you see men afterwards standing in something resembling a line, firing in a regimented and orderly fashion, or even performing bayonet charges does not mean it's "linear tactics" as /u/DonaldFDraper commonly likes to explain :)