Fuel is not supposed to come out of the jet tube on its own. If it does, the float level is a bit too high or the jet set to low. It is the vacuum created by air passing through the carb venturi that "sucks" the fuel out of the jet tube.
However, you should be able to use a flashlight and look down into the jet to see fuel. If you don't, then take a look in the float bowl and make sure there is gas in it. On an HS series carb the bowl should be 1/2 to 2/3 full.
Your spark test is good. Cars do not go badly out of ignition time on their own but points can wear to the point that the spark is weak and timing off just enough that the engine has trouble starting and running. If your car has points, I would go completely through the valve adjustment process and give it an ignition tune up before you spend more than a cursory exam of fuel delivery and float level. Do not touch the mixture until you have gone through the ignition.
Remember that a quick test is to tip a thimble of gasoline down the carb throat and try to start the engine. If it fires, runs briefly, then dies, you have a fuel delivery problem to sort out.