As far as I know, both Dread and Chivalry both increase the Public Order in a city / castle. But from my experience, I know that only Dread decreases the enemy's morale, but I'm not sure what Chivalry does. So, what are all their effects?
medieval-2-total-war – What do Dread and Chivalry Do in Medieval 2: Total War?
medieval-2-total-war
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In most cases it's the repeated purchase or use of units, buildings, and abilities related to that particular guild - for example, frequently buying and using assassins will lead to the Assassin's Guild offering to open a chapter in one of your cities. Which city is chosen is determined by where you do the missions/recruiting/etc - in the above example, if you buy a lot of assassin's from a single city, that city is probably going to be the one that the Assassin's Guild offers to set up in.
The same goes for the second and third level guilds, with some changes to the rules: you can only have one second-level guild of a type in your empire, and there can only be one third-level guild of a type in the whole world. Also note that you can only have one guild in each city - once you accept a guild in a city no other guild will offer to build there unless you destroy the old guild first. You will also gradually lose guild points, meaning that unless you're actively working on a specific guild it's very unlikely you'll ever achieve a third-level structure.
For detailed information on what actions attract which guilds, as well as all the mechanics for them, follow this link: http://www.twcenter.net/forums/showthread.php?t=77577
If assassination is too chivalrous for you, I'd gladly suggest a much better, bolder alternative!
I understand your plight - the Pope is too risky a target for an assassination; risking a star agent and a good friend on the mission is a questionable decision. The inquisitors are annoying, and chasing them down is reactive behaviour; surely there is a better way.
Look at the Papacy. Now look at you. Now look at the Papacy again. Do you wish they were gone, daring to shout insults at you from what is (or will soon be) your own backyard? Make it so!
The Pope tends to keep his faction small; not that he isn't ambitious, rather his neighbors are about as capable as him when it comes to warfare. He probably only has a few regions; he may even be restricted to a single territory. Crush his annoying little state for offending you!
Sometimes, he may have the gall to call a crusade against you. This will rarely happen, and most of the time it will not even be possible - he is only capable of rallying Christendom every few decades, so more often than not, even if he wanted to, he would be unable to cause you more trouble than he's already started.
After ye've ripped off his head and spat down his neck, consider giving his successor a chance to repent for his sins and offer to reconcile with him - you are, after all, a forgiving soul. You're generous too: Give him back the city you took (sack beforehand, he won't care!), maybe even offer him a few of those you captured on the way (suddenly, the larger the Papacy used to be, the better!); don't forget to offer him an alliance so that you can protect him from those who seek to exploit his momentary weakness. More often than not, the new Pope will realize that you're the most faithful believer around, and your troubles will be resolved; if, at some point, he forgets who his best friend is, feel free to repeat the above - most Popes who would fall as far as to excommunicate the most pious nation around have lost God's protection, and you must do the Lord's work and reform the Papacy whenever necessary.
Best Answer
In the Battlefield:
The more dread a general has, the more morale damage his troops do to the enemy's morale. On the other hand, the more Chivalry a general has, the more his troops will refuse to rout.
Essentially you either have your troops be harder to rout, or make it easier for your troops to rout the enemy.
However you can have more than one general in your army, so effectively you can get both effects (although only the leading general [who carries the horn] will receive battle-traits for that specific battle).
In the Campaign Map
Both increase public order, if a General is appointed as Governor (Being in the Settlement). However the faction leader gets relationship penalties/bonuses if he has high dread/chivalry (respectively) towards other factions.
Ideally you want your faction leader/ faction heir to be investing in Chivalry and non-inheriting sons / other generals to be dread or a mixture. That maximizes your relations and combat effectiveness.