You got some good advice from the hydrofoil goat. Wasnt expecting to have to type that.
Are movement dice a worthy investment?
I think so, especially once you start doing 4 on 4 fights: they really help keep track of what units have moved already, and it avoids having to remember what each Mech did.
Is minimum distance and mech heat a thing?
Yes. LRMs for example have a minimum range of 6.
That means that at range 1, you take a 6 to-hit penalty, and then you reduce the penalty for each hex the target is further away. So 5 penalty at 2, 4 penalty at 3. At range 6, the penalty is 1, at range 7, there is no penalty.
Al most all weapons generate heat. An LRM20 creates 6 heats for example. After the weapons and physical phases are done, you total up all heat you generated and received that turn, and then subtract the heat your heatsinks can remove (often: 10, though some 'Mechs can do more. An Awesome can sink 28...!)
How do you deal with a leg being destroyed?
You automatically fall, and the lost leg causes a 5 penalty to any piloting skill rolls after that. You'll need one immediately to avoid MechWarrior damage, and then in future turns you can attempt to stand on your remaining leg, though you'll have that 5 penalty again. Every time you fail to get up, you take more falling damage. Surrendering the 'Mech after it loses a leg often makes sense.
Another thing you can consider doing in such a situation: firing while prone. Possible while you have 2 arms: you push up with both and can then fire all torso weapons, and the weapons of 1 arm of your choice.
Well, also any Head mounted weapons; those are rare.
Where do I go from here? Embrace the beginners rules or add some flavour or play the full rules?
No shame in doing the beginner rules a few times until you're comfortable with them and want to learn more. Information overload is a thing.