I would say want you have on the bottom of the carb is a fuel solenoid. which maybe bad and that's why you have flooding in the engine. have it check out, or the carb look at.
The fuel solenoid does nothing other than to block off the main jet so fuel supply to the engine is cut off.
Flooding is caused by the float not cutting off the fuel to the bowl.
The bowl has a vent to let air in so that the fuel can get out and that is where it is leaking from.
Go to
http://outdoorpowerinfo.com/repairs/ for complete instruction on cleaning the carb, it will be varnished / gummed up.
You will most likely need a rebuild kit and when you get that get an in line fuel tap & fit it as the first thing after the fuel tank so you can then turn it off when the mower is not in use and also to service the filter, pump & carb.
Check the oil level if it is high or smells like fuel then change it, run the mower for 10 minutes and change it again & the filter.
As for the power problem, check the battery voltage, mower off and battery voltage engine running, running must be higher then engine off & must be at least 12.5 and prefferably 13 V or higher.
While in storage the terminals have most likely corroded a little so get a can of contact cleaner, unplug everything, give the terminals a good spray, plug in & unplug a few times.. then another spray & replug.