It depends on the origin and the destination. Their Magic Transit service explicitly allow this, and I assume they have agreements with other AS in case something goes wrong on either side (it often does). You'd have to directly ask them to know specifically but I don't think they would answer since that's proprietary information.
They do routing. Even if you're connecting to a non Cloudflare server, the traffic may still be routed through their servers.
Why would they want to peek traffic? Most likely for statistics (most frequently visited websites etc).
Can you give an example of a BGP route or traceroute to a site not on Clownflare that was routed through Clownflare?
It depends on the origin and the destination. Their Magic Transit service explicitly allow this, and I assume they have agreements with other AS in case something goes wrong on either side (it often does). You'd have to directly ask them to know specifically but I don't think they would answer since that's proprietary information.