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Question about Reaction: it is stated that characters gain their reaction at the start of their turn, but that you can only use 1 reaction per round.

Does that mean that a character that has its initiative at the middle of a round, which is using a reaction gained during the last round will not be able to use its new reaction after its turn during this round but will need to wait the next round to use it ?

This interpretation seems the only possible with the rules as written, but it disadvantage all characters who are not the first one AND THE LAST one in the initiative order (since it is the only situation in which the use of reaction is not restricted since round and turn stack for them), which is really strange.

Thank you for your help.

It's like normal distance, but the first 5ft are free and don't count for the 5-10-5-10 rule. So you have:

5ft: 1 diagonal (consider this free, now we start counting normally)
10ft: 2 diagonals
20ft: 3 diagonals
25ft: 4 diagonals

This way, the wording is consistent.

No it's not, since with this explanation the 10 feet rule would not be an exception, and the second paragraph would not be needed since it creates by its wording a difference with the 10 feet exception.

Is there an official FAQ on the matter ?

If you gave 15ft reach three diagonals, then measuring a range (for example, how far can you shoot or cast a spell) would be different than measuring a reach. So then if you had 15ft of reach you would be able to attack someone who couldn't cast a 15ft range spell back at you. That would be very strange and confusing.

On the other hand, if you treat 10ft reach the same as everyone else, then it becomes really easy to close in on someone with a reach weapon. Because if you come in at the diagonal, you're 15ft away so therefore out of reach, and then move one square and you're 5ft away and you haven't provoked an AoO because you didn't leave a threatened square. In first edition they tried some really convoluted things to solve that, but in the end the easiest was to do the same as in Pathfinder 2.

So the general rule is, to determine the range of your diagonal reach, calculate squares in the same way as for movement or spell/ranged attack range. But for 10ft reach attacks specifically, you can reach two diagonals.

Now, reaching 15ft would also allow you to reach only two diagonals, since it takes 15ft of movement to move two diagonal squares. So at that point the reach radius overlaps. On the horizontal/vertical line though, 15ft reach is three squares and 10ft is two squares as the normal result of using movement cost to determine reach.

So yeah, it's a bit of a kludge, but it's impossible to do everything nicely. Looking at it from a geometry point of view, you're trying to neatly fill a circle with equal-size squares, and if your circle is very small, you get ugly edges.

I'm not sure to understand the true rule here, since it does not fit with shroud explanation, if you admit that 25 feet reach is 4 squares diagonal (rules as written), a 25 reach would be able to hit a spell caster unable to respond with a 25 range spell (3 squares diagonal).

So what is the final rule to follow ?
15 feet is 2 square diagonal or 3 square diagonal ?
25 feet is 3 squares diagonal or 4 squares diagonal ?