Minmatar Completed in 2020 using an Alpha Clone – excludes L4 Agents, Low and Null Sec.

What are COSMOS missions?
COSMOS missions were introduced with the Cold War patch in 2005.
This Dev blog by CCP Capslock provides a good description of how they were meant to function and the benefits available.
Why run a one time mission?
Most pilots ran them for a very fast faction standing boost. The added incentive was the reward, which normally comprised of Storyline modules or Storyline Blue Print Copies. These modules often had slightly better attributes than their Tech I counterparts.
While this seems unimportant in Eve today, back then it provided an edge. Tech II items were in short supply and the blueprints were controlled by a very small number of people. 2006 saw the introduction of invention and a surge in supply of Tech 2 items. Storyline modules have persisted because of their reduced Powergrid and CPU requirements.
They tend to fit when nothing else will.

Rules – There really is only 1 rule for COSMOS missions,
Don’t talk to the Agent unless you have done your research and have the items required !
When you finish a mission the next mission is automatically offered and expires in 7 days.
Almost all COSMOS agents are one time only – forever.
Decline, or Fail, a mission and you don’t get another chance and the rest of the missions will not be accessible – ever.
Agents will ask for items found in COSMOS DED sites. Be warned, these are farmed and you may have trouble getting the items you need – see rule 1.

Access – COSMOS agents will not talk to you unless you have high Faction standing. Standing with their Corporation is not taken into account.
If you are eyeing off the 2 run Tempest Fleet Issue offered by Makor Desto be aware that he won’t talk to you until you have a Minmatar Faction standing of 9.90

Rewards – Faction Standing Gains and Corporation Standing Gains awarded at completion of the three of four mission series. (Beats doing 16 standard missions).
Storyline Modules are awarded on completion and sometimes during the mission series.
There is no LP gain for the Corporations the Agents belong to.

The storyline modules and BPC’s offered when you speak to the Agent seem to be awarded randomly, by that I mean what I receive will differ from what you receive.
I don’t know if doing all the missions gives you access to all loot, but looking at various old mission descriptions, the type of BPC’s and modules are similar. I guess CCP did that so you had to do all the missions rather than just hit up one agent.

List of Resources:
Eve University COSMOS missions – plenty of errors so be careful
Eve – Special Agents Toolkit v1.05 – some issues but reliable.
Fuzzwork Agents in Space – so far this has proved very useful.
The Nosy Gamer – Lore and personal insight from around 2011.