EVE New Citizens Q&A

One chap takes the mission. Rewards are shared evenly amongst everyone in their fleet when he completes mission back at agent (might have to be in system as well I forget).
EXCEPTION : Faction standing gain from storyline missions (every 16th mission will give one). The faction gain will only go to the player that took the mission, other rewards are divided evenly as normal.

This is just mission rewards, loot etc is not part of mission and you will have to divide amongst yourself. Loot required to complete mission will have to be given to mission taker or he cannot complete the mission, You cannot complete someone else's mission.
Bounties from destroyed NPCs will be shared with everyone in the fleet and ON THE GRID of the mission. You will not get anything if you are not actually on the battlefield.
Interesting question about the item, never came up in all these years but I guarantee only the mission taker will get the item reward as otherwise whole alliances would join fleet and instantly generate 10.000+ L4 implants.

Corporate and agent standings, along with mission completion rewards are shared equally among all members of the fleet. Those members do not have to be in the same system, or region, but simply logged on and in the fleet.

It is only faction standings from completing storyline missions that are not shared across the fleet members, since that would make it far too easy to gain standings for alts and the like

Fundamentally:

Direct mission rewards from the agent (the ISK and loyalty points) are either shared out equally among all members of the operating fleet, wherever in the EVE universe they are located, or claimed solely by the person who received the mission from the agent. The mission-runner gets a pop-up when he turns the mission in, asking what he wants to do.

Mission-critical loot must be in the mission-runner's possession when the mission is turned in, either in his/her active ship's cargo hold or in his/her station hangar.

Items offered as rewards are given solely to the pilot dealing directly with the agent. IE: if the reward's an implant, only the person directly assigned the mission gets the implant.

Bounties for destroyed NPC ships are shared out among all fleet members who are "on grid" (within local sensor range, basically) at the time the NPC is killed.

Division of miscellaneous loot and salvage from the mission is basically at the discretion of whoever has the loot in their possession. (Common arrangements may involve selling it all at the local trade hub and splitting the proceeds, or putting the loot into a corporation hangar to be shared out for use by whoever needs it.)