You're way off on PE's.
The Shiller PE is 26* right now. P/E moves inversely with interest rates. With rates around 1% (i.e. driving up P/E's) only distressed companies trade at 5x P/E ratios.
2. "AirBnB takes a 3% cut on transactions"
You overlooked 2/3 of their revenue.
"Airbnb generates revenue by taking a 3% cut of each booking along with a 6% to 12% service fee from guests"
3. "for them to have a $25 billion valuation, that's $5 billion in revenue or $167 billion in bookings"
The relevant data is in the article. Note they made $340 on $2.2B of bookings (about %15 of bookings).
"Airbnb generated $340 million of revenue in the third quarter, on bookings of $2.2 billion". You can just annualize that. Current run rate revenue is ~$1.4B. Current run rate bookings are ~$8.8B.
I don't have an opinion on the valuation but I can understand how they could be worth $25B. Airbnb's incremental cost for processing customers is probably close to zero. If they double bookings again, they'd double top line revenue ~$3B. I could easily see $1B of that flowing to the bottom line.
Also there are network effects/customer captivity at play that bode well for the long term. What do I mean? It would be hard to create a competitor to Airbnb because no one wants to establish their reputation on more that 1 platform (same under-appreciated advantage Ebay has).