1: If a urban barbarian start every combat with his DEX-Rage (Controlled Rage (Ex)), did he get this DEX-bonus to his initiative?
2: Same question, but starting the combat aware of the opponents.
Link to Initiative Rules (d20pfsrd.com)
At the start of a battle, each combatant makes an initiative check. An initiative check is a Dexterity check. Each character applies his or her Dexterity modifier to the roll, as well as other modifiers from feats, spells, and other effects. Characters act in order, counting down from the highest result to the lowest. In every round that follows, the characters act in the same order (unless a character takes an action that results in his or her initiative changing; see Special Initiative Actions).
If two or more combatants have the same initiative check result, the combatants who are tied act in order of total initiative modifier (highest first). If there is still a tie, the tied characters should roll to determine which one of them goes before the other.
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Surprise
When a combat starts, if you are not aware of your opponents and they are aware of you, you're surprised.
Determining Awareness
Sometimes all the combatants on a side are aware of their opponents, sometimes none are, and sometimes only some of them are. Sometimes a few combatants on each side are aware and the other combatants on each side are unaware.
Determining awareness may call for Perception checks or other checks.
The Surprise Round
If some but not all of the combatants are aware of their opponents, a surprise round happens before regular rounds begin. In initiative order (highest to lowest), combatants who started the battle aware of their opponents each take a standard or move action during the surprise round. You can also take free actions during the surprise round. If no one or everyone is surprised, no surprise round occurs.
Unaware Combatants
Combatants who are unaware at the start of battle don't get to act in the surprise round. Unaware combatants are flat-footed because they have not acted yet, so they lose any Dexterity bonus to AC.
3: What with other (spell)buffs which give a bonus to DEX / initiative?
If he is raging when combat starts yes, otherwise no. This will usually mean no. If he is initiating the surprise round yes. Or if his party is going to initiate the surprise round I'd him start to rage in the 'pre-round' before combat starts to get the bonus.
Cats grace spell grants +4 dex, and would apply if it was active when combat started (say through pre-combat buffing cause you know you are about to slam open the door and attack the unaware inhabitants on the other side). Any other spell/ability that gives +dex or +init would work the same.