If an army was to rout, would the soldiers take their armour off to help them flee more easily/not be identified by the winning army?
What kind of clothes would an archer (who as far as I'm aware made up the majority of the armies back then) wear under their livery coats etc?
The answer to this depends entirely on the type of armour they're wearing, but basically boils down to either their ordinary clothing or their underwear. In the case of their underwear, this wasn't a matter of tighty whities or boxers, but also of a linen shirt, which absorbed sweat and body odour and could be washed, while the woolen outer garment could be brushed. The question of whether both outer and under garments or just the under garments were worn with armour is somewhat tricky to answer.
We know from multiple mid-to-late 15th century manuscript illuminations that brigandines and other sleeveless forms of armour were worn over everyday clothing, even if short sleeved mail shirts were sometimes worn under the bridgandine. The tricky part is that, in the early half of the 15th century, long sleeved forms of armour were preferred, so we can't tell whether the archers are wearing their full set of clothing under their textile armour, or if they're just wearing their under garments. Arming scenes for common soldiers are very rare so, while 13th century evidence suggests that the over wear was worn under the textile armour, we can't be sure whether this was the case in the 15th century. It's entirely possible that only the under garments were worn beneath long sleeved textile armour in many cases.
As to the question of whether men would cast aside their armour when fleeing, this is again a question that's hard to answer. The only burial of 15th century soldiers that I'm aware of is from the Battle of Towton in 1461. At least 38 individuals were buried, although only 28 intact skulls were found, and they show a large degree of trauma. For the most part, wounds inflicted on the torso cluster around the extremities, such as the arms, the legs and the neck, which could mean that the torso was relatively protected - there are some cuts to the outer shoulder, clavical and buttocks - or it could mean that attacks on the torso only penetrated soft tissue, leaving no marks on bone.
For my part, I'm inclined to believe that the lack of cuts to the torso is evidence of armour. The skeletons found at Visby, where many men were buried in their armour, show a similar lack of attacks on the torso and a preference for attacking the extremities, albeit the lower legs rather than the upper arms and hands. Meanwhile, the 11th century burials from the Priory of St. Andrew, Fishergate show a much more marked preference for targeting the torso, explicable if they are indeed the fallen Norwegians from 1066.
The problem is that we don't know whether the men of the Towton mass grave were killed in the battle itself, or if they were killed after the rout, as the pursuit phase began. The wounds, which are particularly concentrated in their right hands and arms, are just as likely to have come from attacks on their sword arms while fighting as they are to come from the unarmed right hand being used defensively against a sword stroke. The wounds to the back of the head could have, and in the case of the blunt force wounds likely did, come as the man lay on the ground following blows to the front of the skull in face to face combat. The only real evidence to suggest that these men weren't killed on the battlefield is that the mass grave is some distance from it, but even here we must be cautious, since it's also possible that they were men of the town and were retrieved from the battlefield later or may have been leading elements of the victorious army who were caught up by an unexpected knot of resistance during their pursuit.
There is, however, some evidence that armour would be discarded during flight. In recounting the Scottish defeat at Pinkie Cleugh in 1547, William Patten mentions that the Scottish took off their jacks (jacks of plate, similar to a brigandine, rather than linen jacks) during their flight. This was probably easier when the jack was loosely laced rather than fasted with hooks or long laces, so it might be that only some of the Scots were able to shed their jacks when they initially fled. Some 15th century brigandines were split at the sides, and probably laced there, while others buckled at the front. Neither would have been precisely easy to shuck in the heat of the moment, but as with the Scottish jacks there would likely have been some who managed it, while others wouldn't have to time to during the initial flight. Other archers would have worn only textile armour, but the range of fastening systems was the same, so the same difficulties apply to them as well.
TL:DR
Archers wore their ordinary clothes underneath their armour, and some may have been able to shed their armour when the fled, while others wouldn't be able to undo the fastenings of their armour easily while on the run.
References
