We do enjoy our hinoki tub, although we have had some minor cracking with some black growth on the outside of one of the corner joints. We’ve also found that hair and skin cells seem to stay floating on the surface even though we always shower before entering.
I am sorry you are having some problems with the tub.
About the dust staying afloat, I do not think it is a problem of the wooden tub, although chances are that when you drain the tub, small particles attach to the wood more easily than it would be in the case of an enamel or acrylic tub.
I think you can get rid of these light particles in 2 ways:
1) letting the tub overflow from the top
2) using a thin net to skim the surface of water
About the mold and cracks, please send some pictures, I`d be glad to advice.