Our propane heat furnace is no longer working and needs to be replaced. Will the air conditioner part of a central air/heat still work if we decide to use something other than central heat?

Likely depending on what is wrong with the furnace. They share very few parts, and most of them are cheap to replace. Even the fan uses a different relay and motor windings. If the heat exchanger is severely cracked and would lose too much air, you could have somebody build a tin box to replace it.