I have an archer hero and he needs to get new arrows or repair his bow, how to do that without going back to the demesne?

Once you have the appropriate upgrade in your castle, you can build a "Storehouse" in other provinces. This allows heroes to access the treasury and castle shops, including repair services.

thanks, makes sense but have not reached that yet!

Another one, it seems quite hard to up the mood of the provinces, how you do that? Mine are constantly rebelling, how I can fight that? garrisons?

Rebellion comes from low morale.

Certain evil/abusive deeds in random events will cause low morale (or sometimes doing the good thing to an evil race)
Occasionally you can do an evil sounding thing that actually improves morale. (burn the witch / elf / whatever, confiscate a found treasure and give to the easy to manipulate mortals, etc).

A freshly conquered human territory will have a temporary morale penalty, it will not last long enough to cause rebellion.

Conquering a free non-human colony will cause permanent huge morale penalty there, beware.

Conquering a non human colony that was already conquered by the AI will have no morale penalty (you liberated them).

Garrisoning troops reduces accumulation rate of unrest meaning it takes longer to rebell but they still will rebell if they have a permanent morale modifier. However the garrison kills the rebels so who cares. And worst case scenario they become independent and you can always reconquer them... or let them remain free to reduce corruption (corruption bad! look it up here on forums)

You can also use various shops to repair your equipment. They are randomly spread all over the map and are marked by sign post if you have it enabled, but it takes 1 turn to repair items, as you have to enter them as any other location.

When waging a military campaign, having another hero on hand to run weapons back and forth between the frontline and the nearest shop/castle isn't a bad idea either.

thanks all...

There is no possibility to see the impact an answer to an event will do beforehand (or even after)? If I decide to burn the witch or spare her life for example, how I know what it is doing in the end? Should I write on a piece of paper the mood of each provinces and compare before/after?

As for heroes, it is difficult to have more than 2, so it means one is used as the errand boy?

shops also charge double to repair your items then your own artisans do. You say it like this is an optional thing :P.
Making such supply depo provinces is absolutely essential to every strategy.
Although personally I omit the libraries from most supply depo locations.