It is legal, yes.
Note that move actions are defined one square at a time. The running enemy could choose to stop when you teleport in to threaten them. But if they don’t, and leave a threatened square, they provoke from you.
Actions are not designed to happen at the same time, rather they happen in discrete chunks.
This is clear from the design of turn based games in general, and especially D&D. This is also apparent, as things that grant reactions usually specify specific timing, either before or after its trigger. Example of the Ready Action (PHB p. 193):
To do so, you can take the Ready action on your turn so that you can act later in the round using your reaction…[w]hen the trigger occurs, you can either take your reaction right after the trigger finishes or ignore the trigger.
There are no clearly written rules for resolving multiple readied actions on the same trigger.
However, the easiest, and most straightforward way to resolve this situation is in initiative order for all those readied on the same trigger, and seems to best fit the spirit of the rules, and how they work in general.
An alternative to this, would be to resolve order based on dexterity bonus, however, this introduces additional ties, which should already be resolved by initiative order.
Your last example does let the monster "save" for each attack, but it's still a valid and powerful tactic.
Preventing the monster from responding in between each attack is still a powerful benefit, and may potentially finish it off before it gets another turn.
Being hit while asleep from the Sleep spell, they would wake up on the first strike and lose the unconscious status, and being hit by something that gives a save on damage, such as Tasha's Hideous Laughter, would give them a save for each hit regardless of if all actions happen simultaneously or not, as per the wording of the spells, as each hit is still a separate instance of the target taking damage. Note that victims of Tasha's Hideous Laughter and Sleep would still be prone until they get a turn to stand up.
Finally, the general rule is that things are resolved in initiative order.
There is no specific rule overriding that. Following D&D 5e's system of general rules applying unless a specific one overrides it, I'd conclude this is likely the RAI and RAW approach. See SRD p. 90:
Initiative determines the order of turns during combat. [...] This is the order (called the initiative order) in which they act during each round. The initiative order remains the same from round to round.
Best Answer
Your example is tricky because of how readied actions are worded.
From the description of readied actions (PHB p. 193, emphasis added):
In your example, the enemy with the crossbow would use their reaction to take their readied crossbow shot on the PC's turn, triggered by the PC entering the room. After that attack is finished the PC could use their reaction to take their readied Dodge action. This could give them the benefit of the Dodge action against subsequent attacks, but not the triggering attack.
In order for two readied actions to occur "simultaneously," they would both have to specify the same triggering condition.
From the description of initiative (PHB p. 189, emphasis added):
The clause "the order in which they act" does not say "the order in which they take turns," which implies that initiative order governs the sequence of reactions and other off-turn actions in addition to the turn order.
Thus, if two creatures readied actions and specified the same triggering condition, I would rule that the reactions are resolved in initiative order of the readying creatures upon the occurrence of that trigger.