My wife plays a wizard. We hear Orcs coming up the hall. The wizard decides to ready a spell for when the door opens. So she casts the spell and holds it, until 10 seconds later, the door opens, initiating the trigger. Boom - the fireball goes off. Combat begins. Can she cast a spell on the first round of combat even though she released the energy of the readied spell as a triggered reaction?

My first inclination is, yes, because she didn't cast the spell at the beginning of combat, she did that when she readied the spell. She just released its energy as a reaction to the door opening, outlined on page 193.

No. Readying actions is not how you should handle an ambush.

It is a mistake for the DM to allow players to spend actions (by readying an action to use later) before combat has started. Until you roll for initiative, gameplay isn't broken up into actions in that way.

Instead, the DM should rule that the Orcs are surprised by the players, who have set up an ambush for whoever opens the door. The rules for surprise are written on Page 72 of the Basic Rules and page 189 of the Player's Handbook:

The DM determines who might be surprised. If neither side tries to
be stealthy, they automatically notice each other. Otherwise, the DM
compares the Dexterity (Stealth) checks of anyone hiding with the
passive Wisdom (Perception) score of each creature on the opposing
side. Any character or monster that doesn’t notice a threat is
surprised at the start of the encounter.

If you’re surprised, you can't move or take an action on your first turn of the
combat, and you can't take a reaction until that turn ends. A member of a group can
be surprised even if the other members aren't.

In practice using the regular surprise rules will work out much like allowing the PCs to ready actions before the combat starts. The wizard could cast a fireball on the turn the Orcs spend being surprised, and then another one one the next turn, as the Orcs start to fight. If they roll well enough on initiative, they might get both spells off before the Orcs can do anything at all.

If it makes sense given the situation (or just seems more fun), the DM could probably allow the players to have surprised the Orcs without requiring any Stealth checks. But beware if you're a player and you ask for that, as it might give the DM license to have monsters ambush your party as you go through some future doorway, without allowing you a Perception check either!

The wizard can't Ready a spell before combat begins

The wizard can cast spells during her first round, but she can't release a readied spell before her first round because she can't Ready a spell before combat begins.

As Jeremy Crawford clarified:

The options, including Ready, in the "Actions in Combat" section (PH, 192–93) are meant to be used in combat, after rolling initiative.

This isn't just an arbitrary restriction; it is a rule meant to streamline play. To see why, lets look at the first step in every combat, Surprise:

The GM determines who might be surprised. If neither side tries to be stealthy, they automatically notice each other. [...] Any character or monster that doesn’t notice a threat is surprised at the start of the encounter.

If you're surprised, you can't move or take an action on your first turn of the combat, and you can't take a reaction until that turn ends.

If the Orcs are surprised, they effectively waste their first turn and the wizard still gets to cast her fireball before the Orcs have a chance to do anything. Basically, the game simulates ambushes with the Surprise, not with Ready actions.

If the Orcs are not surprised, then you basically have a standoff that hinges (pun intended) around that door. The Wizard wants to Ready an action and so do the Orcs, but if everyone is going to ready an action for when the door opens, what's the point? Just play through the first round of combat when the door opens.

As explained in Order of Combat:

The game organizes the chaos of combat into a cycle of rounds and turns.

If you allow combats to begin with a bunch of reactions, you effectively reintroduce that chaos for no reason. For example, initiative tells you which turn occurs first, but if combat begins with reactions, whose reaction occurs first?

A readied spell and the spell slot used to cast it are wasted after 6 seconds

This isn't immediately obvious, but it's the result of several rules.

From Order of Combat:

A round represents about 6 seconds in the game world. During a round, each participant in a battle takes a turn.

From Ready:

you can take the Ready action on your turn, which lets you act using your reaction before the start of your next turn.

When you ready a spell, you cast it as normal but hold its energy, which you release with your reaction when the trigger occurs.

From Spell Slots:

When a character casts a spell, he or she expends a slot of that spell's level or higher

Basically, when you Ready a spell, you cast it and expend a spell slot, but you can only release the readied spell before your next turn and the time between turns is 6 seconds.

Yes — everything you say is correct. The only restriction on casting multiple spells on the same turn comes in when you cast a spell as a bonus action. Your reaction is not a bonus action.

Yes

In this case, the reaction starts the combat. Before the reaction, there is no turn. Once combat starts (after the reaction), she gets her turn, at which point she had casted no spells on her turn.