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The Pearl of Power states

You can use an action to speak its command word and regain one expended spell slot. If the expended slot was of 4th level or higher, the new slot is 3rd level.

I'm a level 7 Wizard, with 1 4th-level slot, and 3 3rd-level slots. I use my 4-th level slot, and use the Pearl of Power. Do I now have 4 3rd-level slots? What happens to this newly created slot on a long rest?

It doesn't create new slots, it restores a used slot, sometimes at a lower level

Pearl of Power states:

You can use an action to speak its command word and regain one expended spell slot.

So you can only regain a slot that has already been used. But:

If the expended slot was of 4th level or higher, the new slot is 3rd level.

So, if you choose to regain a slot above level three, then it is effectively temporarily downgraded to level 3. That means, in your example, if you have cast a fourth level spell and no other spells, that temporarily you would have four level three slots and no level four slot.

What happens when you take a long rest?

After you take a long rest, all of your normal spell slots return (in your example, this would be 1 at 4th level and 1 at 3rd level). The extra 3rd-level slot you had created with the Pearl of Power has simply reverted to 4th level - as it was previously. This slot has not been 'lost' as at no point had you created an additional new slot, you'd simply refilled an empty one.

Using the Pearl of Power may have temporarily changed your build slightly but you never gained anything better than what you had in your original build - what you did temporarily gain was always equivalent to, or worse than, what you started with. So there'd be no benefit to this persisting through a long rest, and potentially a considerable cost (i.e. losing a stronger slot in lieu of gaining an extra weaker slot).

The expended 4th level slot reverts to the 4th level slot after a long rest

You didn't create a slot, you refilled an empty one. The key is this text in the item description:

regain one expended spell slot.

The rules for a long rest say that all of your spell slots are restored. This item doesn't change that; if it changed that it would say so.

Per your comment, you cast a 4th level spell and no third level spells.

• A 7th level wizard starts with 4/3/3/1 spell slots for spell levels
  1/2/3/4 respectively.



• You cast Banishment: your available spell slots are now 4/3/3/0




• You use the pearl of power.



• 
  

  You now have 4/3/4*/0 since your 4th level slot was refilled with a 3rd level slot's worth of magic.

  
  
  

  
  

  If the expended slot was of 4th level or higher, the new slot is 3rd
  level. (item, SRD, p. 232)

  
  

  
  

To answer your bolded question - Yes.

Until a long rest is completed you can expend up to 4 3rd level spells ... which you could have done before burning that 4th level slot¹. This temporary condition changes with a long rest - it does not add a slot to your basic allotment. A similar thing happens with Sorcerers and spell points converted to spell slots².

You can illustrate the pearl's benefit as "the 4th level slot, filled to the 3/4 line" like an empty ale mug being 3/4ths refilled. You expend one more 3rd level slot of any prepared spell, not another 4th level slot, until you finish a long rest -- which could have done if you'd used a 4th level slot to cast a 3rd level spell rather than banishment.

To check consistency revert to a 5th level wizard. You have 1 3rd level slot. You cast fly. Your 3rd level spell slot is empty until you refilled during a long rest. Use the pearl of power, refilling that slot before the long rest. You can cast fly again before the next long rest since your 3rd level spot was refilled.

¹Casting a Spell at a Higher Level

_{When a spellcaster casts a spell using a slot that is of a higher
level than the spell, the spell assumes the higher level for that
casting. For instance, if Umara casts magic missile using one of her
2nd-level slots, that magic missile is 2nd level. Effectively, the
spell expands to fill the slot it is put into. Some spells, such as
magic missile and cure wounds, have more powerful effects when cast at
a higher level, as detailed in a spell’s description.}

² Creating Spell Slots.

_{You can transform unexpended sorcery points into one spell slot as a bonus action on your turn. The Creating Spell Slots table shows the cost of creating a spell slot of a given level. You can create spell slots no higher in level than 5th. Any spell slot you create with this feature vanishes when you finish a long rest. (SRD, p. 43)}

Your intuition is correct - you will end up with an "extra" 3rd-level slot. You target your expended 4th-level slot, and the Pearl tries to fill the whole thing, but only has enough power to equal a 3rd-level slot.

This question is especially relevant when a Warlock is involved, as Warlocks with 4th or 5th-level slots don't even have any 3rd-level slots by default - but by using a Pearl of Power, they can end up with one.

Keep in mind, you can only regain an expended spell slot with a Pearl of Power, so you can't use it right when you wake up in the morning to get an extra spell slot. You'll have to actually cast a spell with that slot first. Also, if you expend a 1st-level spell slot, and then use the Pearl on that slot, you will only get a 1st-level slot back. Same goes for 2nd-level slots; you can't use the Pearl of Power to increase the size of your spell slots, just to replace expended ones.