D&D 5E battle rules rework.I used this site
https://roll20.net/compendium/dnd5e/BookIndex to make sure I'm not bending any rules unintentionally. In overall I would say that while some of your changes are logical some others mitigated differences between classes way too much.
1. Actions and attacks.1.1. Disengage should be separated from jumps. And it should be full action for everyone but lvl2+ rogue.
It's way too OP that we can just always jump away from every threat on every character and kills positioning tactics. And aside from being faithfull to D&D 5E, it's logical, you need some agility to easily move away from a fight. Disengage being bonus actions for rogues is part of them being agile and fast.
1.2. Please, add Dodge action.
Sometimes it feels more important to do everything possible to save the character than do anything else with him (hello, Gale), but we simply can't do that.
1.3. Please, make Help work as in D&D 5E.
Working exactly like in D&D 5E it'll still help in a way against prone (in preventive way even), sleeping, enwebbed, ensnared and some other things. It will lose anti-downed effect ofcourse, but that one is unrealistic and useless, we are getting our characters up with help just to get them downed again in a turn.
1.4. Hide should be full action for everyone but lvl2+ rouge.
Same as with Disengage rogues are pro with hiding others are not. Come on, you're killing appealing sides of my favourite class, Larian.
1.5. Please, add Ready action or mode.
It's inconvenient that on each turn we are forced to act now or never, with no way to decide to wait a bit for enemy's move.
I can imagine 3 ways of implementing Ready action from D&D 5E:
A) Right by the book. Click Ready to switch on planning mode, click on enemy we are going to observe, click on direction (let's add segments around the enemy) or a point on landscape or one of our characters to say if we are waiting for movement or being on the spot or attack attempt, click on our action that should be used, click Ready again to switch off planning mode and move on, consume Reaction point and lock the character from being used. If the enemy does what we expected - execute prepared actions consuming Action, Bonus action and movement points, if not - skip the turn. Maybe better would be to change the rules here slightly and allow our character to act at the end of turn if the readied reaction wasn't executed.
B) D:OS1 approach. Click on Ready to move our character to the end of queue for current turn. After the end of turn move our character back where he was originally by initiative.
C) Semi-D:OS1 approach. Click on Ready to make our character able to make his turn at any given moment by the end of current turn.
1.6. Please, add Search action to represent more active search by forcing one more perception roll.
In some areas it's way obvious that there is something hidden and in table-top session in such area I would insist on searching more thoroughly than just take one look, one roll and move on. Plus, it'll make players reload less to find secrets. To make things more interesting and avoid abusing of active search let's make failing to many rolls for same spot potentially dangerous. For example a trap my be activated acidently, while searching.
1.7. Shove should be full action/type of attack as in D&D 5E. Maybe with melee attack being done in addion at the same time, if melee weapon chosen at the moment of shoving.
Bonus actions stand for something fast and simple or doable at the same time as normal action. Some classes ability to perform some full actions as bonus actions is way of showing their mastery in corresponding areas. Effectivly shoving someone, who doesn't want to be shoved, in a middle of a fight is not that fast and simple even with some training. Only with a melee atack it maybe possible to shove and strike with weapon at the same time.
1.8. Please-please, add Grapple action/type of attack right as in D&D 5E plus living shield effect.
While on first sight it's not needed, cause you just control one enemy, while loosing your mobility and there are so many other ways to controll an enemy already… on the second sight it's a cool way to save a character by using grappled enemy as a living shield. So technicly grappling someone should slow down our character, but incoming attacks should be mostly transferred to the grappled enemy, and his sapient comrades should think twice before attacking our grappling character.
1.9. Please, make it possible to perform additional bonus actions instead of full actions.
According to D&D 5E we can perform what would be bonus action more than once in a turn, just with full action(s) being replaced by additional bonus action(s). It's not that hard to jump twice in a few seconds for example. And it would be great improvement to our movement possibilities in a fight.
2. Spellcasting.2.1. We should have a way to intentionally stop concentrating on spells and cantrips with prolonged effects that require concentration.
It's very inconvinient and inconsistent when we cast fog cloud for example and then can't do a thing in that area, while it's our own magic, on which we are concentrating to keep it existing. By D&D 5E a caster can just stop concentrating at any moment for free to get rid of his own spell effect.
2.2. If there is nothing to put on fire/or freeze to ice and the spell or cantrip doesn't create fire/ice surface by D&D5e - don't create fire/ice surface.
While it's fun to put things on fire (pyromaniac here), and fights were fun in D:OS2 with all those surfaces, with D&D lack of teleportation abilities for melee fighters too many surfaces turn each fight into mages, archers and jumpers contest. In D:OS1 there was same exact problem, I'm playing it now with 2 melee classes and it's painfull as hell even with full backpacks of grenades to deal with surfaces.
2.3. Please, tie scrolls and magic items to classes as they are supposed to be by D&D 5E.
Having them being usable by everyone with no restrictions doesn't make the game more fun, but makes it harder to tell one class from another and decide who should carry them. And after that it removes some tactical planing considering characters firepower.