Will there still be enough viable yeast in solution to ensure carbonization?
sure - unless you filter it out, there will still be enough. bear in mind, you will need to let the bottled beer sit at room temp in order to carbonate though.
No, you will need to add yeast back to the beer. 2-3 days at 38 should drop most all of the yeast out of suspension, this works great for filtering out yeast and "clearing" the beer. If you bottle from this vessel you could rouse the yeast back into suspension while mixing in the sugar, but I don't know how well this would work at 38 degrees.
I have never had a beer not carbonate after cold crashing added yeast or no. There is still plenty of yeast in suspension after 48 hours cold time.