[RPG] apply Expertise to a skill proficiency granted by background or racial

Yes.

Proficiency isn't tied to anything except your character level (i.e if you multiclass for example Rogue 3 / Cleric 2, your character level is 5, which means your proficiency bonus is +3).

Your proficiency bonus increases everything you're proficient in, full stop. This applies to attacks with weapons you're proficient with, saving throws you're proficient in, skills you're proficient in, etc.

From the Basic Rules, p. 7:

Proficiency Bonus

The table that appears in your class description shows your proficiency bonus, which is +2 for a 1st-level character. Your proficiency bonus applies to many of the numbers you’ll be recording on your character sheet:

• Attack rolls using weapons you’re proficient with
• Attack rolls with spells you cast
• Ability checks using skills you’re proficient in
• Ability checks using tools you’re proficient with
• Saving throws you’re proficient in
• Saving throw DCs for spells you cast (explained in each spellcasting class)

Yes, Expertise affects passive Perception checks.

Passive Checks (PHB 175)

Here's how to determine a character's total for a passive check:

10 + all modifiers that normally apply to the check [...]

For example, if a 1st-level character has a Wisdom of 15 and proficiency in Perception, he or she has a passive Wisdom (Perception) score of 14.

Simply put, if you have a WIS mod of +3 and a proficiency bonus of +4 (with proficiency in Perception), your passive Perception score will be 10 + 7, or 17. If you have Expertise in Perception, the +4 is doubled, so you're correct in judging that your passive Perception score will be 21.

Even better, if you're in a context that would grant you advantage on your Perception check, your passive Perception would be no less than 26, thanks to how advantage modifies your passive Perception score:

If the character has advantage on the check, add 5. For disadvantage, subtract 5.