1) The Masters chosen by the Grail are always three members of the Tohsaka, Matou and Einzbern. The other four are usually magi with a desire for the Holy Grail and thus obtain the Command Seals. It is not uncommon for someone without magical talent to be a Master, but I suppose that the Master needs at least a connection to magecraft. Urdu Ryuunosuke was the descendant of a family of magi.
2) Servants are Heroic Spirits, and Divine Spirits are in another category albeit they are both spirits. Divine Spirits can be Heroes like the centaur Chiron, but to be summoned as Servants they would have to discard a degree of their divinity and power. Heracles ascended to God after his death, but he is summoned as a Demigod. Chiron discards his centaur form to not get so obvious. Fictional heroes aren't possible, they can only if at this point of time it isn't known whether or not they existed. For example, if Goku was a legend of thousands of years ago and is known fairly well, even if he didn't exist, human thoughts will give him form.
3) It's a battle royale, but basic rules are not spilling the secrets of the War to any outsiders and silencing any witness. Seven Masters and Seven Servants only. The Church chooses the overseer, but there may be a special occasion where a Servant class called Ruler is the overseer. He or she will interfere if the Holy Grail War is endangered and possesses two Command Seals for every Servant, thus 14.
4) The Grail is divided in Lesser Grail and Greater Grail. The first has the souls of six Servants and can grant a wish, but not so omnipotent. It has so much mana it can 'give a push' towards a goal, but not something like ruling the Galaxy or silly things. The Gretaer Grail is activated with seven souls of Servants and opens a path to Akasha, the Root of the World, which is the dream of all magi given how they want the wellspring of knowledge Akashic Records. The Greater Grail activation is the true purpose of the War initially thought by the Three Families.
5) Depending on the Mana rank or Independent Action of the Servant, it can last in the world for some days. The Servant can form a bond with another Master to anchor themselves in the present, but if the Master has another Servant it wouldn't make sense since only one pair can win. Also, sustaining two Servants is impossible unless you are some sort or Praha furnace. Caster of Fate Stay Night has essentially no Master and provides mana for herself, and Kuzuki only has a contract to keep her materialised.
6) Command Seals are on the level of True Magic and are essential. When used, if the Servant agrees with the order they will receive a boost in power to accomplish it, if not they will weaken. If the order is too broad like 'obey me' it won't work at all unless the Master is an excellent magus. If the order is 'defeat the enemy' and the Servant obeys, it will receive much power able to boost stats or for a devastating attack. An order like 'kill yourself' is pretty much absolute. Command Seals can be slightly disobeyed only with a truly strong will, albeit barely and not enough to break the order, or with a Magic Resistance of A-Rank.
7) The Grail needs at least six souls to activate its functions, and it also has a sort of mind on its own. It won't accept Masters who have been holing up like cowards and not proved themselves worthy of a wish. Only a pair can win.
) The rules imposed by the Grail. Even such artifact can't summon more than seven Heroes, and it also uses prana to aid the Masters with their prana cost with Servants. Heroic Spirits are such heavy and powerful beings that they usually can't be maintained in the World by humans, but the Grail helps. With two Servants, even with the Grail helping, a Master would remain sick all the time while nerfing the Servants.
9) The War was set to be in Fuyuki only. Well. There have been 726 claims of the Holy Grail appearance throughout history, but almost none or maybe none at all were Wars with a wish at stake. Anyway, the Grail is fixed to appear in one of the four spiritual locations of Fuyuki, so the War takes place only in the city. In Fate Apocrypha though, someone managed to 'steal' the Grail at the end of the Third War, taking it to Romania for a War with 7 Black Servants and 7 Red Servants.
10) One can fight anytime and anywhere unless you cause too much of a commotion. Beams of light and explosions aren't exactly easy to hide, especially during the day. At night it's easier, and Masters can easily use magecraft to avoid discovery. Master can be killed anytime by anyone, it's a sound strategy to kill the Masters so that the Servant disappears. Even so, their Servant will not permit it. Also, depending on the alignment of the Hero, killing an enemy Master wouldn't be appreciated an may ruin the relationship with his Master.
11) It gathers energy for 60 years to create the Command Seals and have enough prana to bring forth souls of Heroic Spirits. It has a consciousness, but it was tainted during the Third War from an irregular Servant called Avenger. Said Servant was a simple farmer who was used as a scapegoat to contain All The Evils In The World and was called Angra Mainyu like the evil deity, even if the boy wasn't him. As a Servant he was pathetically weak and was killed first. His soul entering the Grail, he tainted it with his curses and now the Grail can only grant wishes through destruction.
12) Fluids are excellent conductors of prana. If a Master doesn't have enough naturally to grant his Servant enough power, he can have sexual intercourse to temporarily offer a decent amount. Well, it may be an excuse to bang legendary heroes. How would you feel to have the chance of doing it with Jeanne d'Arc, Atalanta or Circe?
13) The Grail doesn't have so much strength that it can summon the true Heroes. So it copies the souls stored in the Throne of Heroes and put them in seven vessels called Classes according to the skill set of the hero in his life. It is not rare for a Heroic Spirit to have qualifications for many classes: Heracles can be summoned as any class apart from Caster, and Cu Chulainn can be a Lancer, Rider, Caster, Berserker. Seven vessels exist and seven classes also, so only one Archer, one Saber and so forth.
14) You can go to the overseer of the War, give up the Command Spells and break the contract with your Servant if he hasn't died already, which is the usual reason one forfeits. Servants can not leave Fuyuki as it is already a miracle they can exist in the modern world, but they can remain only in Fuyuki or the influence of the Graik would not reach them.Also, the Servants have wishes on their own and it is not necessarily living once again. In fact, few of them have such a wish. There is no time limit, but usually the War lasts two of three weeks.