How well would a hydronic furnace work with a gas tankless water heater?
You can use a tankless boiler with your air handler. The fact that the heat is being distributed using forced air doesn't really affect how you generate the heat, and you should be able to retain most of the benefits of a tankless (i.e. you are not constantly holding a bunch of water at a high temperature) in this configuration.

If you add a tankless heater for domestic hot water, there will not be much use for your electric tank. There is only one use I can imagine: positioning the electric heater after the tankless (in terms of water flow) so that when you turn on the hot water tap you initially draw from the electric tank. This is done sometimes in order to supply hot water faster, as tankless water heaters can require some time to warm up. This sort of application generally calls for a very small water heater, so if your current water heater is designed to serve your entire home,using it with a tankless would probably be very inefficient.

One option you might consider is a combined heat/hot water tankless. I have a Navien CH-240, which provides both heat (via radiators) and hot water. It's small, quiet, and way less expensive to operate than the old oil boiler it replaced. The equipment cost of a combined heat/HW tankless will probably be less than two individual tankless units.