As a tip to everyone, which I'm assuming at least a few people know already, you can earn keys through sweeping instances. So the trick is to farm one key, and then do your wanted missions, since wanted will always give you two keys no matter what. With this method you can end up with 11 free keys a day instead of 10.

Amounts to same thing really. Get yourself an odd number of keys from sweeping (stopping before 10 of course), and then do wanteds to fini* out. Results in 11 keys, plus the free try I guess so 12 free pulls total per day as you said. Saves you on the stam pots.

See earlier comments about how to get free keys during the event. The easiest way (provided your high enough level, which I think is 60) is to just do wanted missions, as that gives you 2 keys per mission, to a maximum of 10. Further juggling around with sweeping instances can get you 11 if you're clever, but never more than 11 free keys a day (plus the 1 free draw every 24 hours).

On the other hand, if you don't do wanted missions because you're trying to level freeze, then well that's your decision. You're probably gonna end up getting exp from trying to sweep for 10 keys than you would from doing wanted missions.