As usual for such questions, let's assume that by "medieval army" you mean specifically an army typical of the Western European High Middle Ages, even more specifically an English, French or Imperial army of the 13th to the 15th century. (I don't know why most people just forget that the Middle Age period one thousand years long and covers vastly different cultures. And that's the European Middle Age period.)
Let's take a famous medieval battle: the battle of Agincourt, which took place in 1415 (at the very end of the Middle Ages). There is a detailed article on Wikipedia, and, even better, there is an entire book, The History of the Battle of Agincourt by Nicholas Harris, London, 1832, available from Archive.org in multiple formats.
How close can they realistically be? Can one army see the other? If they were to charge against each other, how long would would it take from the attack order until the moment swords clash?
They would certainly see each other: in the Middle Ages they didn't have over-the-horizon radar and indirect artillery fire. If they cannot see each other there is no way to find each other in order to do battle.
A cavalry horse covers 340 to 400 meters per minute at a gallop. (This is about half the top speed of a race horse.) The armies will most likely be separated by about one third to one half a mile at the start of the battle; when one of the sides charges they move at a trot to about 100-150 meters and then they go to a gallop. How much time will pass between the start of the battle and the first clash depends on the specific tactics.
Swords clash: not likely. The most usual use of a sword on a medieval battle field was for an officer to brandish it crying Forward! After me! Swords were by and large decorative side arms for rich people. Lances clash with the shields, pikes clash with the horses, shields clash against shields...
What is life like on both camps? What do soldiers do while they're not in combat?
They stand guard, forage, train, eat, wash, mend their equipment, dig latrines, sleep, gamble...
Is it believable that those armies fight more than once (They have a round of combat, retreat and sometime later resume fighting)? How many time would pass between rounds?
It depends what you mean by "those armies". Those kings, yes, possibly. Those exact armies, no, not likely. One side wins and one side loses, or one side decides to disengage (a tactical loss) in order to gain a better position elsewhere. Battles are not sporting events, they have a purpose subordinated to the strategic goals. What may happen is that the battle continues over several days; in this case, at night everybody just stays put with minimal combat.
When and for how long do they fight? If night falls, both of them retreat? Or do they keep fighting?
See point 3. They most certainly don't fight at night -- it's the Middle Ages, no way to see in the dark. It may well be the case that both sides stay put and wait for daylight, or one or both sides may decide to disengage, or (less likely) to maneuver (medieval armies were not really able to maneuver tactically).
At any given time during a fight, are ALL of the soldiers fighting? Or some of them remain on camp as a backup?
It's not likely that all the soldiers are fighting, unless (a) the commander has seen a tactical opportunity and has committed all the reserves, or (b) the situation is desperate. A wise commander will keep a force as a reserve, to enable him to exploit any tactical opening. But they won't remain "in camp". They will be on or near the battle field, at the ready.
How big can the armies reasonably be?
Look up the Hundred Years' War on Wikipedia; there is a list of battles, and for each battle you will find a nice table summarizing it. Basically, 5000 men is a sizeable army, 10000 men is a large army, 25000 is a huge army the likes of which the world had never seen before. (The logistics prevailing in the Middle Ages were such that feeding 25000 men for more than a few days was a superhuman feat.)
Note that armies very much prefer not to fight other armies. Usually, a set piece battle takes place either because one side has outmaneuvered the other so that they cannot continue their campaign unless they go through the opposing army, or because both sides stumbled into each other by chance. (In the case of the battle of Agincourt, the first case happended; the defending French outmaneuvered the English, putting them in a position where they had to go through the much larger French army if they wanted to avoid death by starvation.) If at all possible, the commander of the attacking side will try to go around the defending army, and the commander of the defending side will try to occupy a strong defensive position where the enemy would be strongly incentivised to refuse to attack.
This being the Middle Ages, it is possible for two armies to do battle although any commander in the Antiquity or the Modern Age would have preferred to avoid it; they were not fully rational in the Middle Ages, and they did have to operate with very limited tactical flexibility, given the generally poor training of the soldiers. Even the knights had poor military training -- they were well trained in fighting as individuals, but had only minimal notions of fighting in formation, obeying orders, maintaining unit cohesion and so on.