You can do a pressure test of the cooling system to see if there is a leak path letting coolant into the exhaust.
When running the engine try to determine if the exhaust flow out the tailpipe changes with engine rpm. If the flow out doesn't go up with rpms you probably have a restriction somewhere.
Depending on how long you ran the engine some white smoke on initial running is not that unussual. There are a lot of things that get burned off as the engine warms up and a lot of them have contents that will create white smoke. As long as the coolant and oil levels remain at acceptable levels run it and see how things change with run time.
