[RPG] Does a controlled mount share its rider’s turn

Mounted combat can get a little confusing but most of the complexity falls away if you remember that the mount and the rider remain 2 separate creatures and the only things that change is exactly what it says on p. 198.

Mounted or not you and your mount have:

• 1 move
• 1 action
• 1 bonus action if circumstances permit
• 1 reaction if circumstances permit

When you mount (and nothing allows you to change this short of dismounting and remounting - which unless you are in melee is trivially accomplished) you decide to:

1. Allow the mount to act independently. It can do any or all of the first 3 of these on its initiative count each time its turn comes up. It can choose any of the 10 available actions that are available to it (e.g. a non-spellcasting mount can't cast a spell).

2. Control the mount. It then acts on your initiative and can still do any or all of the first 3 of these but it is limited to (Dash, Disengage or Dodge) for its action. If the mount had a bonus action it could do whatever that was in addition to its action.

In either case, what you do affects you and what your mount does affects it. Specifically, if your mount:

• Dashes: it can move as its movement and move again as its Dash, you go along for the ride.
• Disengages: it can move without provoking Attacks of Opportunity, you go along for the ride and do not provoke Attacks of Opportunity because you are not moving.
• Dodges: it can move and gets the AC and saving throw benefit, you go along for the ride and ... don't. However, you can choose to Dodge as your action or as a bonus action if you have that capability.

Importantly, you can move. For a normal mount like a horse, this would normally involve dismounting as the first part of your move but for a big creature like a sandworm you could move without dismounting and your movement would provoke Attacks of Opportunity (exception Disengage) which could not target your mount. This is different to your mount's movement which provoke Attacks of Opportunity which can target you or your mount.

Yes, it can

The relevant rules for opportunity attacks (from the Basic Rules):

You can make an opportunity attack when a hostile creature that you can see moves out of your reach. To make the opportunity attack, you use your reaction to make one melee attack against the provoking creature. The attack interrupts the provoking creature’s movement, occurring right before the creature leaves your reach.

As you've quoted, controlling a mount does not place any restrictions on the mount's freedom to use its Reaction.

But as a DM, I probably wouldn't allow it

I interpret a controlled mount to be fully under the control of the controlling character. A mount that is free to react as it chooses is by definition not under control. If the rider wishes to instruct it how to react, that would use the rider's own reaction, and so the rider would have to forgo her own opportunity attack.