Do my HVAC ducts need to hang away from floor joists?
You should consider switching to low profile hvac duct.  You can get a wider version that is shorter but carries the same air.  If you can't find something off the shelf an hvac professional can design/fabricate one that will work.  Won't hurt you to get a quote and see what your hvac guy can do for you.  Would be a shame to put the effort into finishing a basement and have crazy bulkheads running everywhere.