Hi ALi,
Yes, this all sounds quite normal, so don't panic. I kept all my nails through chemo but there were two main effects... discolouration and what are called 'Beau lines' (love the irony there) which are the horizontal ridges that correspond to doses of chemo.
When I had Taxotere I was told to paint my nails with dark nail varnish as a means of protection, and I think to a degree it acted as glue, holding them together as layers started to peel off as the nails grew through. If you are up for a it, you might find that nail varnish (can be clear now, the dark is to do with UV senstivity) will help to keep the nails stuck together as they grow. You need to go right to the edges, and I was advised to just overlap the cuticles. It worked for me.
Keeping your nails short as they grow through is good, but be careful not to risk any hangnails, as the risk of infection is best avoided.
My finger nails grew through very quickly, but my big toes took a year to completely grow out all the yukky yellowness.
Sounds like you're doing the right things, so keep on as you are.