in my opinion, u do not have to tie them too tightly. just tight enuf for the base of the mushroom to touch the rock without it floating away. it should take around 1-2weeks for it to attatched. i tried this with xenia and it took 5 days to stick to the rock. try not to touch and disturb them during this period. they are stressed out from being fragged. yes the original stem will regen into a new mushroom. ignore the curly white stuff. as long as it is not melting, it will be fine. mushrooms are seriously resilient.
the string of black stuff it is spitting out is either waste or its own zooxanthellae. if it has 2 mouths, it may split on its own. but this may take awhile. no worries and again as long as it is not melting, i would not get too worried. corals spit out "black" stuff all the time. they do that when they finish digesting a meal, or when u change them to different lighting positions.