Can someone steal the coins now that the letter of guarantee is made public?
No.
Or I just lose anonymity?
Yes.
I did NOT know what the url was for checking my order.
For what it's worth: I couldn't find how to use the Order number to check your transaction, until NowYouAreSee posted the link. That's something BestMixer can improve on, an entry field at the top of the page would work.
why doesn't bestmixer make the system idiot proof so that it's impossible for me to proceed WITHOUT downloading the guarantee?
The Letter of Guarantee is the one thing that tells you you're using the real website. Keep a copy, check the signed message, and confirm it's signed from the correct address.