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DesertSnow

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In every characters data, you can see his/her conviction in the faction he/she belongs (a number in brackets). I think that certain titles can shift their allegiance (eg "tribune of the soldiers" to the military).
 

Cheexsta

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What DesertSnow said, except as of 2.1 minor titles no longer give conviction by themselves; they only give conviction for that job if the character had it as an objective.

For example, a character with the ambition to be a Military Tribune gets +2 Military Conviction if he is given that title. A character wanting to be a Legate gets +5, etc.
 

Cheexsta

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Not as far as I can tell, they don't.

Having an ambition to be governor gives them +6 mercantile conviction when they become one, though.

The +0.5 per month thing only applies if you un-comment the relevant lines (ie remove the # symbol) in static_modifiers.txt.