So "line formations" came about in response to the downfall of "pike and shot armies." Pike and Shot armies dominated most of the early modern era (about 1453-1815). Basically large groups of pikemen were mixed in with groups of musketeers, and those formations made up the infantry component of your standard European army.

However as the pike declined in usefulness and muskets became more accurate and more deadly, commanders began to add more and more musketeers to replace pikemen. The main purpose of the pikemen were to protect the musketeers against cavalry. So when the bayonet gets invented around the late 1600s. Pikemen became obsolete and European armies began to get rid of pikemen entirely. At the same time musket technology was improving as well. Paper cartridges and flintlock technology drastically increased fire rates.

So European military thinkers now had to answer two questions. One, how do we maximize the firepower of our troops? And two, how do we protect them from cavalry. The answer was line formations. The long line formations allowed as more soldiers to fire at once, and despite all the technological advances, muskets still weren't super accurate meaning that more muskets firing at once= greater chance of your army actually hitting something. As for the second question about cavalry, well the advent of the bayonet meant that musketeers weren't totally helpless when confronted with cavalry/melee situations, so European armies could afford to stretch out their lines.

So it wasn't like everyone gathered together to agree on those rules, its just that line formations were the most effective way of organizing troops at that time.

how did the soldiers endure psychological pressure knowing they're going to die when waiting the enemy to shoot

So that's actually another reason armies used line formations. See it was easier to drill and keep soldiers clam by having them very close to their fellow soldiers. The thinking was that a soldier in a line formation was far less likely to run when his comrades were beside him/behind him.

So just to add that these things don't develop in bubble, and line tactics were practiced during the pike and shot era, they didn't just happen overnight. Armies during the Thirty Years War, experimented with line tactics. Early military theorists like Maurice of Nassau developed early versions of line tactics and focused heavily on using more musketeers than pikemen.

Well, the transition to line is more of efficiency rather than "gentleman's agreement" since the weapons were inaccurate. The start comes from the Thirty Years War where Gustavus Aldophus tried using lines instead of squares of pikes with infantry at the edges. By the Beginning of the Eighteenth Century, most Western armies were using lines to make the musket efficient. Over time it would grow beyond just lines to skirmishes and by the late Eighteenth Century, there was much discussion about whether infantry should march in a line or column formation.

As for soldiers, it's a complicated thing. First, muskets are not instantly deadly as our contemporary weapons, it's very possible for soldiers to receive a multitude of injuries (the French Marshal Oudinot took over thirty injuries over his service, all from various times of service when he wasn't able to receive the best medical service). Musket balls are also not very good at penetrating compare to modern bullets, there are even stories of French cavalry soldiers shaking off balls that hits them through the battle and got caught in the folds of their jacket. So a soldier wasn't going to their death, but at best a welt from a ball (since the quality of the powder, which is not always the best, could slow down a ball as well).

The life of a soldier during this time period was terrible though. The Line works as a unit, no one can be slow or fast because they would break it and caus disunity of fire power. This was done with tireless drilling, and if a soldier didn't go along, they'd be beat and punished with lashes or The Gauntlet in order to force the mindless drill into their mind. By the time they are an effective unit, there should be little thought in a soldiers mind of what he needs to do, which is just to obey orders and March with his unit.