It's backfiring when cold because the mixture is too lean & it's lean when it's cold because the manifold & other parts have not warmed up enough to convert the liquid droplets of fuel from the carb into a gas.

It's the gas that burns remember, not the liquid which essentially is passing through the cylinders unscathed.

The engine is cutting out when you stab the throttle because the accelerator pump isn't working, which is not uncommon.
When you stab the throttle you give the engine a gutfull of air, particularly through this bigger bore carb.

Unfortunately, the engine doesn't run on air, so it stops running, but when you close the throttle you re-establish a better airflow through the carb, it draws fuel from the discharge nozzle which makes its way to the cylinders & it runs again.
Is the choke function fully operational & correctly adjusted? My money say it isn't.

Sounds to me like you would benefit from the purchase of a carb overhaul kit & having it installed by someone who knows what he is doing. This should be followed up by a proper carb tune & rejetting as required as it seems to me that someone has fitted the first tired old carb that came along & expected it to work.

It won't

If you get smoke from the electrics, it won't be related to the carb in any way. [you do have a proper flame arresting air filter on it, don't you?] Electrical wiring smoke has a nasty acrid smell to it that takes your breath away, so you'll know it when you smell it.