Question 1: That bit of text is probably there to cover those Heroes that do start out with the Reputation Advantage, as well as letting players and GMs know that said Advantage doesn't have to be the only means for a Hero to earn Reputation.

Question 2: What that sidebar is saying is that a Hero (or Villain) can't declare their approach for the round in an Action Sequence as simply "I dodge!"  Dodging or otherwise avoiding damage is a reactive thing, and this game wants the Heroes to be more proactive during Action Sequences.  You can still spend Raises to reduce Wounds taken from an attack or consequence, but the sidebar is trying to communicate that 'dodging' isn't going to be the only thing your Hero should be doing.

Question 3: Given that Skills are not inherently linked to specific Traits, it's quite possible for a canny player to find plenty uses for Resolve.  It can be used with Brawling or Weaponry to account for how the Hero is gradually wearing down their foe's defenses, using their stamina to simply outlast their opponent, either literally wearing them out or being patient enough to wait for their foe to get frustrated before striking.  It can be used with Athletics or Ride to account for the Hero's stamina in outracing pursuit, or their single-minded determination not let their quarry get away.  It can be used with Sailing to account for how the Hero doesn't so much avoid the effects of a bad storm while aiding in various ship-based tasks but rather simply endures what the sea throws at her.  Use it with Convince to draw out a verbal debate to the point that the other person throws up their hands in frustration that you're so resolute in your stance on an issue.  One of the great things about this system is that with a little creative thinking, you can combine Traits and Skills in some fun and unusual ways that other RPGs wouldn't allow simply because they've dogmatically tied certain skills to specific ability scores.

Question 4: Most bribes for non-nobles are going to be fairly trivial, to the poiint that it's not really going to matter, and is instead going to be accounted for in the Tempt skill roll; at most it might count as Flair and grant the Hero a bonus die.  For nobles, you can probably just generalize what the spent Wealth Points mean, though a good description might warrant bonus dice for Flair, and mechanically it'd let you re-roll one die on a social Risk as per the bottom of page 164.

Question 5: Contested actions are simply a form of Risk, and you compare the number of Raises the Hero spent on the action to the number of Raises the opposition has to spend.  For non-Villain opponents, you can probalby set a threshold similar to a consquence, or just use the Strength of a Brute Squad.  So with the example given of a Hero trying to bar a door to keep a Strength 5 Brute Squad from breaking through, you'd compare how many Raises the Hero got on their skill roll (we'll use Resolve+Athletics for this one) versus the Brute Squad's Strength.  The Hero gets 3 Raises, which means that 3 of the Brutes are kept from getting past the door, but 2 of them manage to make their way inside (exact description of how would be up to the GM, as the Brute Squad is effectively a consequence in this set up), leaving the Hero with a Strength 2 Brute Squad to deal with.

Question 6: It's possible, but the GM is free to spend Danger Points to alter the scene, or to give Captain Reis what amounts to plot immunity.  So presuming Reis doesn't cut you to ribbons after that first pistol shot, the GM could spend a Danger Point upon Reis taking that 4th Dramatic Wound and have them falling over the railing and into the sea, with the body naught to be found, and then some suitable time later have Reis reappear, fully healed and with a massive grudge against the Heroes.  Also bear in mind that what's fair for the goose is fair for the gander, and it's quite possible that a Villain could employ the same tactic on a Hero.

Also, one thing to bear in mind is that John Wick is generally of the "small press indie gamer" mindset when it comes to the rules, and the book was written to give GMs a fair amount of flexibility in how to interpret things, preferably in favor of whatever makes the game the most fun for the GM and for the players.

This is emphatically not D&D or Pathfinder where the authors expect you as the GM to follow generally strict guidelines in terms of the rules; the current versions aren't as bad as 3.5 D&D with its "rules for everything and everything has a rule" mindset, but they do expect you to conform to a certain standard.

So there's probably going to be a lot of areas in 7th Sea where things aren't explicitly laid out or spelled out for the GM.  And that's not a bad thing, as it means you've got freedom as the GM to interpret things in the way that will be the most fun for your table.

Question 1:

So, do we begin with a Reputation (or many if you purchase the Advantage) or not?

It's an Advantage if you want to START with a Reputation.  Otherwise, the GM can award it to you through play.  

Question 2

There is no mechanic description whatsoever. If I can just 'magic-dodge' anything, why would I spend Raises to mitigate received damages? Do I have to roll something? Does it even take a Raise/Turn to be done?

You can spend a raise to counter a point of damage.  That's the dodge mechanic.  But you don't "dodge".  All of this is just semantics to break players from the "I hit you," "I avoid you hit" cycle.  They want you do describe it, embellish it, and otherwise make it sound cool.  But mechanically, you spend a Raise to cause a Wound ("I hit!") and a Raise to counter the Wound ("No you don't!").

Question 3

Why would I ever care about the Trait 'Resolve'?

Sure.  You go ahead and use Resolve as a dump stat.  That's cute.  Now watch me pile on the pressure and start calling for Resolve+Skill rolls.  I can EASILY see Resolve being used in dozens of social Risks, and not a few physical ones.  Just because you don't need it for the death spiral doesn't mean you don't need it.

Question 4

Wealth points.

Eh.  Do you want wealth accumulation to be a big goal of the heroes?  If not, this works fine.  Those minor bribes?  Just hand wave it.  You slip the bartender a few guilders and move on.  You want to buy the house a round of drinks?  Ok, that costs 1 Wealth point.  You want to bribe the Duke of Charouse?  Yeah, that's at least 1 Wealth.  You need a new sword because the BBG took yours?  1 Wealth.  It's very abstract, but it does work.

Rather than messing with fractions, just multiply everything by 10. I'd HIGHLY recommend this for pirate games where things like shares and carousing are issues.

Question 5

I can't find any mechanics about Contested actions. If I try to push open a door and someone else tries to keep it closed at the same time, how do I resolve that?

It really comes down to this: whoever wants something more bids more raises.  If someone every bids ALL of their raises, they win; unless both parties bid ALL of their Raises.  Note that if a Hero (or Villain) bids ALL of his Raises on a Goal, the Villain (or Hero) is free to spend his Raises on other things (like messing with the opposition).

Question 6

Is it just me or I can really kill Captain Reis in 1 Round if I bring 4 pistols and roll 4 Raises?

Nope.  First, it's Captain Reis, so f**k that.  Second, Guard brutes, because no.  Third, hello Syrneth force field, because it's f**king Captain Reis.  Fourth, I'm going to spend a Danger Point on that to make Raises cost 15 because...you know.  You know what, let's make that 2 danger points and Captain Reis' first raise to cause Pressure, because...again.  So now you have 2 Raises and have to spend 1 of those to overcome pressure, only to have a brute jump in front of your bullet.  Now pardon me while Reis spends a Raise to slice that brace of psitols off your chest, another to kick it over to a brute squad, and a third to command her men to set you on fire with a syrneth super soaker (see Daughter of Fate novel for details).  On the next round, first consequence is the brute squad with your three remaining pistols pitches them over the side.  I'm still gonna spend a DP to knock raises to 15.  And when you get that second shot off, I'll spend a Raise to either soak that wound with my Syrneth force field (I'm sorry, you can't find that in the books?  It's Captain f**k'n Reis!) or have another of my loyal brutes take a bullet for me – just to keep you guessing.  Oh, and the duelist?  He's still pissed because I spent a raise to cleave his sword in two with my syrneth crescent blade.

So no.  You can't kill Captain Reis with 4 pistols and 4 Raises.  Not if the GM is doing his/her job.

(I could also be a real jerk and just keep slicing the pistols in half each time you spend a raise to pull one.)

(Sorry, I don't mean to come off as snide with that response.  Yes, all things being equal you should be able to do 4 DW with 4 guns and 4 raises.  But villains don't play fair and neither should you.  Being conservative, Reis should be a 20 dice villain (10 Strength/10 Influence), and should on average be sitting on10 Raises in a round PLUS the crew, including a half dozen+ minor villains in the mix.  In that scenario, good luck getting off 4 shots.  And while the Syrneth force field isn't specifically canon, I'd use it anyway because it's Captain Reis!  You don't get to 10 influence letting pirates take shots at you.  So if your players are going to take a run at Reis because they've analyzed the game system and believe they've found a weakness, they deserve what they get.)

edit: if you want to be "fair," that syrneth force field needs a Danger Point to activate it.  But I'd just rule it counters a Dramatic Wound with a Raise because its Captain Reis.  ;)