Few more notes.
1) Under the rules of engagement, the way you write it implies that "Engage" is something a player will be familiar with. Remember that you have invented that concept in this document, so you need to go into a bit more.
I agree with you. I also have to write an "extended actual play" example to make it more comprehensible. This will probably come soon, but right now one of my goals was to keep this PDF limited to 2 pages, to make it a short read, and also to avoid feature-creep that leads to bloat.
2) Not a critique, just my note. I think OA will actually be much more common in this game vs a normal combat. Main reason is in normal combat you can often "engage" 3 things at the same time. I can run over to the area with the level, beat up the 2 orcs, and pull the level...etc. In this one, since you can only engage with 1 thing at a time, I think there will be more pressure to switch engagements...therefore drawing OAs. Again, that actually might be a good thing.
No sweat, that's exactly the kind of feedback that improves these rules, be they harsh or not!
About the commonality of Opportunity Attacks... well... maybe I didn't make it clear enough, but the intention is that if you can engage a creature and beat it, you are still free to move and engage another one. If you can kill creatures with one attack, and have enough extra attacks, you can basically wipe the battlefield without taking a single Opportunity Attack.
1. In your example, you would be able to engage one orc and beat it, then engage the second orc and beat it, then engage the lever and pull it. All in the same round, if you can kill the orcs fast enough.
2. Or alternatively: you move directly to the lever in order to engage it, but is intercepted by an orc. You know that you could attempt to engage the lever anyway and take an OA, or you could use your Action to engage the lever cleanly... but you're probably going to the intercepted by the second orc whichever option you choose. So you decide to simply attack. On the orcs turn, one is already engaged and attacks; the other engages you and also attacks. The next round, you use your Action to engage the lever and activate it (the orcs are engaged and can't intercept any longer). You're now engaged with both orcs and the lever.
3. Or even: you see two orcs protecting a lever, they are ready to interrupt you. You use your Action to Dash into the lever, engage and activate it. (The unengaged orcs though, taking advantage of your desperation to use the lever, move and engage the cleric that stayed behind... and you can't intercept them. It's a trade off with stakes that change accordingly to the situation at hand).
3) Movement Speed. I think you imply most of the these, but its important to spell it out.
a) All racial speed bonuses are gone. So dwaves and wood elves have the same speed. Should wood elves be given something for their trouble?
b) What happens if racial and class bonuses are combined? Wood Elf monk? Or a barb/monk? Do these go into "movement advantage"?
Indeed, racial speed bonuses are gone, and this is by design. Racial bonuses to speed are usually "tiny", and this implementation of the rules make the bonuses to speed much more considerable and perceptible (Dash becomes a Bonus Action). Making concessions in the rules to include these "tiny" differences was not worth it (added complexity for little gain) and that's why they are ignored. And since they are ignored, they don't stack with Class Abilities, as in your Wood Elf Monk example.
Something else is also implied, but important: Class Abilities don't stack if you multiclass. So your Wood Elf Monk/Barbarian/Rogue only ever gets one Dash as a Bonus Action... which a Haste Spell or Potion of Speed can improve to "Dash once per turn as part of your movement" until the effect ends.
4) Reach Weapons: Went from no value to OP. Basically I am getting at least 1 free attack almost every combat against medium creatures. That's a hell of an advantage.
That was also my fear, that it would be OP. But you must remember that you only get the "free attack" if a creature engages you, and if you're unengaged. During playtest, two things happened: monsters would avoid long pointy things, and you don't get a "free attack" if YOU engage the creature... or they would engage anyway, the player would get the "free attack", and that was it... he would stay engaged until the end of the combat with the current creature, or with additional creatures that engaged him while he was already engaged.
So I stopped worrying and added the rule.
Of course, any DM is free to limit the rule to monsters only, or remove it completely.
If you ever playtest this as well, I'd honestly really like your feedback.
5) Area of Effect. I feel like 1 or 2 more examples is good here. That section took me several rereads to really get, another good example can make a world of difference. Might a recommend an radius around the caster as an example?
Yep, I agree.
In practice, a radius around the caster would work exactly as the Burning Hands example: The spell range is 0 feet (Self) so it only affects Near creatures. Supposing a radius of 20 feet (circle), you check the table and see that it can affect up to 4 creatures (radius ÷ 5). So you choose 4 Near creatures at will and blast away.
Actually, this method is described in the DMG 249.
6) Have work out if certain abilities need rewrites with this system.
a) Cleave?
b) The ranger's attack all enemies adjacent to it.
Just for a few starters.
I'd rule that the ranger can attack all engaged enemies, and it averages to the same.
In grid combat, if enemies are close together, you can get adjacent to them easily and unleash the attacks... but if they are far apart, you're toast. In these rules, it takes more effort to engage many enemies (unless they engage you), but you can choose to actively use your Action to force engagement, independent of the enemy position (as long as it's Near). Both systems have their pros and cons.