It's highly unlikely creatures need rest to regain their uses of their spell-like abilities; such a restriction would be stated along with such abilities. Otherwise,

The Game Doesn't Say...

Similar to what you want to know is in the Rules Compendium's new rules for Daily-use Items which says

Many magic items have a limited number of uses per day. When the last of these uses is used, that power is no longer available for the day, but the item is still magical and might have other powers available. Unless otherwise noted in an item’s description, any item that has daily uses regains all those uses at dawn each day. (86)

Absent other rules, this seems a good enough house rule as any to manage when a creature regains its uses of its abilities per day.

...But the Sage Opined

In the Dragon #338 Sage Advice column "Official Answers to Your Questions," then-Sage Andy Collins was asked

What exactly does "once per day" mean? Does it mean "once per 24-hour period" or is it recharged after the character rests for 8 hours, like spells?

To which Collins replied

It means that in any given day, the ability may be used once. So what constitutes a day? That's where things get a little tricky and rely on the DM's common senses.

The Sage advises using daybreak as the start of a "day," meaning that all daily-use abilities are recharged in full each morning, regardless of exactly when they were used during the previous 24-hour period. For the vast majority of games, that's when most abilities are recharged anyway (spells, for example), which makes it easy to use.

If your players try to abuse this flexibility--such as by adventuring through the night and then suddenly getting all their abilities back when the sun comes up--the DM should feel free to be more restrictive. It's entirely reasonable to require a full 8 hours of rest before allowing daily-use abilities to recharge (even for those characters who don't require sleep). (84)

That's probably as "official" as it's going to get.

No

When a creature's spell-like abilities are listed as used at a specific caster level--such as 12 in the efreeti's case--that caster level is fixed and usually unaffected by changes in Hit Dice or the addition of templates and class levels.¹

If a creature's spell-like abilities lack a specified caster level, the caster level for those spell-like abilities is the creature's Hit Dice.

1. As with everything in D&D 3.5, there are always exceptions. For example, from page 44 of Dungeon #116 there's this feat:

   Practiced Magic

   [untyped]

   Your spell-like abilities are more powerful.

   Prerequisite: Spellcraft 4 ranks, spell-like abilities.

   Benefit: Your caster level for you spell-like abilities increases by +4. Ths can’t increase your caster level for your spell-like abilities beyond your Hit Dice. However, even if you can’t benefit from the full bonus immediately, if you later gain non-caster-level Hit Dice you may be able to apply the rest of the bonus.

   If you have spellcasting ability from multiple sources (from gained templates or from a character class) you must choose which set of spell-like abilities gain the feat’s effect.

   This does not affect how often you can use your spell-like abilities in a day. It only increases your caster level, which would help you penetrate spell resistance, increase the duration and damage of some spells, and so on.

   Special: You may select this feat multiple times. Each time you choose it, you must apply it to a different set of spell-like abilities.