No, the drop rate actually is that believably bad. I went nearly 30 FATEs in Central Thanalan before getting any.Guessing you are not doing the fates with the correct minion out for the zone you are in. (Each minion has 3 zones which its legendary medallion will drop in if it is out for the fate.)
Uhh, no. The only reason your minion will disappear is if you call out your chocobo. It also temporarily disappears if you mount, but I don't think this actually affects anything because when you get off your mount your minion will reappear.i'm pretty sure whenever the legendary medal procs, your minion will despawn and you need to take it out again.