RAW the Rogue does receive the benefits of Expertise to his passive.
A passive check is a special kind of ability check that
doesn’t involve any die rolls. Such a check can represent
the average result for a task done repeatedly, such as
searching for secret doors over and over again, or can
be used when the DM wants to secretly determine
whether the characters succeed at something without
rolling dice, such as noticing a hidden monster.
Here’s how to determine a character’s total for a
passive check: 10 + all modifiers that normally apply to the check. - Player's Handbook P.174
The player has invested in making this character good at this, this is not a bad thing.
Rogues in general are supposed to be good at this type of thing, hence the proficiency bonus class feature and the player specifically spent a feat on making his passive perception better. The player could've spent this on another feat to specialize in another area (or gain a combat ability). This is a good thing, while it may make it difficult for the Rogue to be surprised by a trap or an ambush you should in no way try to outmaneuver this.
Don't worry, there are downsides to the Rogue for this.
To reliably spot traps and ambushes the rogue will need to be at the head of the marching order so that he has clear sight lines. Rogue's are not particularly hardy and the party as a whole is inviting more risk for this reward. Likewise while he is extremely excellent at spotting physical dangers, the Rogue will not be able to detect magic wards and other dangerous enchantments and may equally blunder into them.
Ways to handle his detection of a trap or an ambush
You are correct in assuming that only that PC has seen the trap/enemy. Unless he has some mode of telepathic communication he will need to speak out and announce the threat to everyone. The best way to handle this sort of thing is to pass notes or send text messages to the player(s) able to see/detect the issue and leave it to them to react and tell someone else. Intelligent enemies will see/hear the rogue warning the party and the combat should start immediately (no surprise round though).
The PC does not become omniscient of the trap upon detecting it.
The PHB itself is very, very vague on what information is received when a PC detects a trap. However, the DMG does have a nice little section about traps, their detection, and disarming them:
If the adventurers detect a trap before
triggering it, they might be able to disarm it, either
permanently or long enough to move past it. You might
call for an Intelligence (Investigation) check for a
character to deduce what needs to be done, followed
by a Dexterity check using thieves' tools to perform the
necessary sabotage...
...In most cases, a trap's description is clear enough
that you can adjudicate whether a character's actions
locate or foil the trap. As with many situations, you
shouldn't allow die rolling to override clever play and
good planning...
...Foiling traps can be a little more complicated.
Consider a trapped treasure chest. If the chest is opened
without first pulling on the two handles set in its sides, a
mechanism inside fires a hail of poison needles toward
anyone in front of it. After inspecting the chest and
making a few checks, the characters are still unsure
if it's trapped. Rather than simply open the chest, they
prop a shield in front of it and push the chest open at
a distance with an iron rod. In this case, the trap still
triggers, but the hail of needles fires harmlessly into
the shield. - Dungeon Master's Guide p. 121
Essentially though whether trap disarming is simply a dex check, a series of checks, and/or involves serious RP is up to you as a DM. I would encourage the open-ended approach the book suggests as it adds complexity and makes trap checking and disarming a more engaging process. Sitdown with the party OOC and discuss what the table as a whole thinks should happen for trap checks and move forward based on that consensus.
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.
Best Answer
What A Familiar Can Do
The sprite familiar can do anything a Sprite enemy/NPC can do acording to their stat block. If a familiar has something like Blind-sense (bat) or better perception than you, you can ask/tell it to look around for you.
The stats block only list bonuses that are proficient, other scores (like Arcana or Religion [+2]) would be determined by the typical way that is done for a player. The Monster Manual lists the mods after the stat, and you'll see they follow the same rules as players (+1 to modifier for every 2 points after over 10, -1 for every 2 points under 2). With the sprite, that means s/he could do a athletics check, but his/her modifier is -4. Yes, you're reading the stats right, are extra bonuses.
It can, even, perform the "help" action for you or a party member for any action the Sprite is able to do. It might even make sense to help your sprite look around (i.e. Advantage with the higher perception modification). Likewise it is terrible at lifting (athletics), but it can still "help" you lift providing you advantage with your strength.
The familiar can provide advantage on attacks to an enemy if he/she is within 5ft, and spends its action "helping" by distracting the enemy (which it can do while invisible) or by flanking rules (which it has to be visible to do).
The familiar can also be dismissed and brought back in a different form (bat, cat, etc.) following the rules in the Find Familiar Spell.
The familiar can, of course, deliver touch spells cast by the Warlock.
What A Familiar Can't Do
The stat block doesn't list any tool proficiency, so the sprite can't pick locks or like. Which means that it can't provide "help" for those tasks either.
As a Warlock, your sprite can attack but at the cost of an action, but unless you can't act for some reason, an Eldritch Blast or other spell is going to be more effective.