Do you unlock anything for the single player campaign when playing multiplayer Mass Effect 3 or vice versa, or are the modes completely separated?
In Mass Effect 3;
Single Player: Has no affect on your Multiplayer besides you gaining experience with different weapons that would ultimately be harder to get in Multiplayer (i.e. Black Widow, and other weapons of N7 Quality.) Allowing you to know what weapons you want with what mods and classes.
Multiplayer: - Has a small but noticeable Affect on your Single Player Campaign.
Whenever you have a successful mission in Multiplayer, you will gain a Galactic Readiness Percentage increase in your Singleplayer (i.e. 5% increase to Galatic Readiness). Every player starts with a minimum Galactic Readiness of 50% (your Galactic Readiness will decay over time if you do not continue to increase it) although there are 6 sectors that make up the "Galactic" Readiness.
Each sector has a map associated with it in Multiplayer which allows you to increase the Sectors Galactic Readiness by selecting and completing a multiplayer mission on that map. Although if you select random map, in multiplayer, and win you will achieve an increase for all the sectors Galactic Readiness. Another thing to keep in mind is the harder difficulty of Multiplayer mission that you successfully complete (i.e. Bronze, Silver, and Gold) the higher percent of increase to your Galactic Readiness you will receive.
Your Galactic Readiness affects your war assets in single player completely. Say for example you have 1000 War Assets and only 50% Galactic Readiness, then only 500 of your War Assets will go towards your goal of achieving a victorious ending in single player. Completing a multiplayer mission while having a readiness rating of 100% grants a 5% boost in experience earned.
Another benefit to single player from multiplayer is the option of adding War Assets to your Single Player Game by Promoting a level 20 character that you have created. This will give you 75 War Assets and this can be done as many times as you wish. This promotion also adds 10 points to your N7 Rating and allows you to start over in Multiplayer on that character.
Doing harder difficulties in Multiplayer doesn't affect your war assets when you promote your characters. so doing gold in hopes of getting more war assets then doing bronze will only disappoint you.
Some proof for you. Hope this is what your looking for, couldn't find anything else.
The single player aspect has no effect on the multiplayer.
Multiplayer, however, can affect your singleplayer, through galactic readyness. As you play multiplayer the galactic readyness has a chance of going up after every match.
Also, once you reach level 20 with one of your character classes in multiplayer, you can "Promote" this character to your single player galactic war effort, granting you 75 points of military strength, and 10 N7 ranks (N7 Ranks has no effect in single player). This will also effectively reset the entire class (For example, soldier, and all it's races included) back to level 1.