ASW – Alternative hero leveling.

1) Machine Lore. Devotion to Machines, and from it, the catapult. To put it mildly, the entire branch is questionable. Will there be any balance changes?

Synergies: Devotion to Machines + rate of fire, Devotion to Machines + tent, Devotion to Machines + catapult - and all of them are Rune-based.

2) Seals. Against Darkness in Light, against Light - in Darkness. There is already Eternal Light against Darkness in Light, is a second perk needed?

There are few perks in Light. This perk will give an additional bonus.

3) School of Light. I don't see a connection between Ritual and Wrath and Teleport.

Punishing and acceleration are very strong under Wrath. Therefore, it will cost more mana.
The Ritual of Light itself is aesthetically pleasing because light is more suitable for restoring good than other magic.
Light is important for the Knight, but there may not be enough mana.

4) School of Darkness. From Master of Pain - Weakening Strike, and from Master of Weakness - Misty Veil? Where is the logic, or are the icons simply mixed up? I don't see a connection between Deadly Misfortune and its branches.

What is the point of Weakening Strike if there is mass weakening?
Weakening reduces damage, and so does Misty Veil.
Deadly Misfortune reduces a parameter, hence Darkness. Darkness is suitable for Necromancers - it's okay. Necromancers have Banshee's Cry. Although it would be more suitable for the Lord of Curses, giving it there is dangerous - the Lord of Pain and the Lord of Mind must be in different branches, and the Fallen One with Pain - in one. Although it is possible to swap the Fallen One with Deadly Misfortune.

5) School of Chaos. The connection between Cold Steel and Lightning is not visible. It is unclear why the more powerful Deadly Cold comes before the Lord of Ice.

Lightning and ice shards stun, albeit in different ways. The connection comes from a semi-physical ability, which is not so critical (and there is nowhere else to put it).

6) School of Summoning. Elemental Equilibrium as a purely passable perk instead of a counter? It is unclear why the Lord of Life comes after Cursed Lands.

Passable, but important. And it is easier for the Knight and other non-summoners to find a counter. It is aesthetically important to achieve elemental equilibrium in order to command them.
Cursed Lands, at least, somewhat restrict movement. In order to raise the dead, no one should be standing on them or have died. And if there are few hit points, it is dangerous to approach - you will die before you reach them. And putting it in another branch is worse.

7) Attack. I would like to see the overpowered BB (Berserk Barbarian) as a level 3 perk, not level 1.

At level 3, they are already very strong.

8) Leadership. Resilience doesn't seem to fit, as it is a bonus to HP, and this is traditionally in defense (by analogy with BB).

A bonus to HP gives resilience to the army, as does HP bonus. Leadership and the army are one and the same. Or should HP bonus also be moved?
+HP is not a direct increase in defense.

9) Education. There is no connection between Economy and Revelation.

Economy also brings experience, and Revelation also brings gold.