[RPG] How does the disengage action work against a monster with the reactive heads ability, such as a hydra

Reactions are actions that you take out of turn (or possibly on your turn) that happen in response to something else. A reaction can never occur without a trigger.

A Ready action is a special type of action that uses your action to prepare for a triggering event, and when that event occurs you use your reaction to perform whatever you had readied.

To step back a moment, you have four "action types" you can make:

• Action
• Bonus Action
• Reaction
• Movement

On each round, you get one of each of these things. On your turn, you can use an action, a bonus action (if some class feature, feat, spell, or some other thing grants you one), and you can move up to your movement speed (if you use the Dash action you also move as your action). You also get a free "interact with an object" (does not include magic item activation/use). Whenever a triggering event for a reaction occurs, you can also use your reaction. Anything that requires a reaction will tell you what the triggering event is, except in the case of a Ready action, where you actually determine the triggering event yourself.

A Ready action uses an action on your turn to, as I mentioned previously, prepare for some triggered event which then uses your reaction to perform. So when you Ready you might say, "I Ready my firebolt cantrip to cast at the first orc that steps through that doorway."

This uses your action for that turn. Then, when the triggering event occurs, you can do as you have described, using your reaction to complete the Ready action, OR you have the option to move up to your movement you instead (irrespective of if you moved or not on your turn).

So, in essence, a Ready action uses both the action and reaction granted for that combat round, and once your Ready action has triggered you cannot act again until your next turn.

One other note for readying spells -- when you Ready a spell you actually cast the spell using your regular turn action, and "hold" it until the triggering event occurs. Note that this usage requires concentration. You can cast the spell as long as you continue concentrating on holding it.

It is important to note also that because you only get one reaction per combat round that if you have taken an Opportunity Attack or cast a spell with a casting time of reaction (such as Shield) then you cannot use your reaction to complete your ready action (but you might still have movement remaining). On the other hand, if you have taken the Ready action and your trigger has occurred and thus you've used your reaction for that round, then you cannot cast a reaction spell or take any Opportunity Attacks (barring any special abilities/feats that might grant additional OA's).

Your rogue can absolutely attack (action), disengage/dash/etc. (bonus action from Rogue Cunning Action) and Uncanny Dodge (reaction) on a single turn, but he can only Uncanny Dodge in reaction to a triggering event (outlined in the rules for Uncanny Dodge).

You already bolded the answer.

Beginning at 5th level, you can attack twice, instead of once, whenever you take the Attack action on your turn.

If you're not taking the Attack action on your turn, then you can't use Extra Attack. Taking the Attack action doesn't just mean attacking - it's a specific action you can perform, listed under Actions in Combat.