1- About the +1 on dice rolls on some skills that some advantages gives (ex: +1 on notice with University); if you roll a natural 10 (and therefore a total of 11), what becomes of that extra 1? Is it wasted or does it count as an additional dice that rolled 1 (i.e. can I 'transfer' that 1 to another group of dice to try and get a raise?).

Yes it becomes an 11, and no the extra 1 doesn't carry over. The threshold to build raises can be bumped up by the GM using Danger Points (to sets of 15 instead of sets of 10,) and that's where the bonus applies on a natural 10 will help you. In other cases, you got a 10, nice...and you've drastically increased the probablity that your other dice will form more Raises.

2- About the rule that allows to 'reroll any single dice' when a skill is at least 3. Would you say that the reroll automaticaly replace your lowest dice or do you get to chose if you use it or not (that is, if the reroll is lower than the original roll).

Personally, I've dropped that rule (along with the other Skill Bonuses and Flair) completely (the JWP presents crew has more or less done so as well and they've replaced it with a "Spend a HP to make your 10s explode." instead as part of Khitai.) If you choose to use the rule though, Player gets to choose. You don't have to re-roll if you don't want to, just like you don't have to make your dice explode if you don't want to either.

If you are rerolling a die, you are physically picking it up and rerolling it.

So if you pick up the low die, it replaces that one.  If you pick up a high die...well, that would be a less than optimal plan.