Nuts are OK to serve in moderation. Walnuts (except black walnuts I think) are safe. Many do not offer peanuts due to the potential of aflatoxin, but if you do offer, they should be human grade and roasted.
Egg is also OK - once again in moderation. Personally I do not feed the shell when pured in the food processor or blender and added to other foods (such as bird bread, etc).
There really isn't anything you can do about when a bird goes into molt, provided that there is not some type of nutritional deficiency causing an early molt.
A change in diet will change droppings, but any change in droppings should be monitored carefully. Because birds metabolism is so high, if a change is because of something they ate, it shows up very quickly...it wouldn't take a couple of weeks. Here are two informative links on the subject (one is a sticky posted at the top of Diet forum).http://www.mytoos.com/forum/ubbthreads.php?ubb=showflat&Number=222099#Post222099http://www.exoticpetvet.net/avian/droppings.html
ETA: Here is a good resource on nuts: Nutrition in a Nutshell
. Also looks like black walnuts are OK too!