I'm using Chaeto in my fuge as well as in my display tank. If I remember right, mine did something similar, but i just kind of attributed it to system shock since I didn't acclimate it or anything like you would with more sensitive life. Honestly, I wouldn't worry to much about it though cause I've only managed to kill a little bit of it by letting part of it dry out when my fuge broke and had to be drained while I fixed it (I just cut that part out and froze it to be sure it was dead before i tossed it).
Whats really interesting is that I've had colonies of it springing up in my display where a tiny piece has broken off of the main bunch and gotten caught in the rockwork. Some of these colonies are in incredibly low light areas but so far it keeps growing. The only difference is that it's a much much deeper shade of green. I also wouldn't worry to much about the "little white things", mine had those when I got it and still does. I've never quite figured out what they are, but I assume they are probably some kind of micro dusters, pods, sponges or something.
Btw, don't know where you got the idea that it doesn't like Flo, she's always been Chaeto's favorite waitress at IHOP. All kidding aside, my understanding is that most people grow chaeto in high current parts of their fuge or sump where it's encouraged to tumble in the water column. I've heard they do this for reasons ranging from "It does this in nature" to "It has to tumble or the parts on the bottom don't get any light and die!" My belief/limited exp with it suggests that it doesn't HAVE to tumble unless your growing it in a huge colony and the bottem is so shaded it gets NO light. I recently saw a pic of a 55g that was used as a chaeto fuge and only held a 2 foot ball of chaeto that was kept spinning to get light at the bottom. The really crazy part was that it was divided in half like every two weeks or something like that. I'm of the opinion though that most people tumble it because it seems to stay in more of a "ball" form if it's tumbled. Mine is not tumbled in the least anymore and it's a freaking serpents nest with parts actually growing up and out of the water.
Sorry for the long winded post, hope this helps. Just relax, take a deep breath, and remember, chaeto's a nitrate sink because it's hard to kill, but like all things, it can take time to adjust to an environment.