Does copying a wizard spell into my spellbook require a prepared spell slot?

Question

Say I am a wizard and I can have a maximum of 3 prepared spells. If I want to copy a spell, say from a scroll, does it count as one of my prepared spells? So for example, if I have 3/3 prepared spells, can I still copy a new spell from a scroll?

Answer 1

No. Prepared and known spells are different for wizards.

Read through the "Spellcasting" section for the Wizard in the PHB (p. 114). In particular:

You prepare the list of wizard spells that are available for you to cast. To do so. choose a number of wizard spells from your spellbook equal to your Intelligence modifier + your wizard level (minimum of one spell). The spells must be of a level for which you have spell slots.

If a wizard can have three prepared spells, they choose three spells from their spellbook. The spellbook may have many more spells than that in it.

The only requirements for copying a new spell into a spellbook are:

When you find a wizard spell of 1st level or higher, you can add it to your spellbook if it is of a level for which you have spell slots and if you can spare the time to decipher and copy it.

...

For each level of the spell, the process takes 2 hours and costs 50 gp.

No mention is made of this newly-copied spell counting as one of the wizard's prepared spells.

But, of course, a wizard who copies a spell into their spellbook doesn't have that spell prepared to cast. In order to be able to cast a newly-copied spell, they must prepare it as any other spell:

You can change your list of prepared spells when you finish a long rest. Preparing a new list of wizard spells requires time spent studying your spellbook and memorizing the incantations and gestures you must make to cast the spell: at least 1 minute per spell level for each spell on your list.

So, overall, to go from a spell on a scroll to having the spell prepared to cast, the wizard must:

1. Copy the spell — 2 hours per level of the spell.
2. Take a long rest — 8 hours.
3. Prepare the new spell — 1 minute per level of the spell.

Answer 2

Yes, you can still copy the spell, assuming you can cast spells of that level.

Per "Copying a Spell into the Book", Wizard class, PHB,

When you find a wizard spell of 1st level or higher, you can add it to your spellbook if it is of a level for which you have spell slots and if you can spare the time to decipher and copy it.

Emphasis mine. You do not need to have a "free slot" available to copy the spell. You only need to be able to cast spells of the same spell level as the spell you are trying to copy.