Why is fuel leaking out out the air intake

Check for an air leak in the fuel line from the tank.

Theres only three basic things that make an engine run.... spark, air and fuel
It starts, therefore you have spark
It runs briefly, you may not have enough air, check your air filter/intake
It then dies, you may have a fuel restriction or are losing your fuel prime, a leak or a bad seal and its sucking air into the fuel system

I doubt a leaking plenum would cause it not to start. Usually a leaking plenum would cause an air leak and make the engine run too fast.
Any trouble codes stored ? Will it start using carb cleaner spray ?

It's not your battery. It might be your IAC valve (intake air control valve) or an EGR valve, maybe even a vacuum leak. The battery is fine. Your fuel filter is most likely fine. When the fuel filter plugs, the car will usualy idle ok but will lose power during acceleration.

Sounds like you have a vacuum leak. When the engine is cold the leak is there but as the engine warms up the parts start to expand thus closing the gaps and sealing the leak. I had this problem on my personal vehicle and it turned out to be lower intake gaskets.

Many times when oil appears in your air intake and in the air filter this is usually caused by a plugged pcv valve Positive crankcase ventilation valve it costs anywhere from 3-10 dollars.As far as it not starting my guess is they checked the air filter and left a vacuum hose or airbox open.Some of these vehicle will barely run if an air leak is present at the airbox containing the air filter.Would double check for any vacuum lines that may have been disconnected around the air box and that the box is shut and sealed properly.

Too much air is entering the inlet manifold making the air/fuel mix too lean to ignite. Can you hear any suspect whistling or hissing from any of the pipes etc whilst the engine is turning over. Check every junction in the air ducting from the air filter box though to the throttle intake. There should be no disconnected pipes or holes. Likewise all the pipes and tubes that connect with the inlet manifold should have secure junctions, the pipes should be free of splits etc and the other ends of these pipes should also be secure to the devices to which they are attached. Be diligent in checking every bit. Two approaches that might make the leak detection quicker: 1) Disconnect the vacuum tubes (those that make connection to the inlet manifold) one at a time and in turn to block the now open pipe to the inlet manifold with your finger. Do this to each tube in turn whilst trying to start the engine, eventually you will find the faulty tube because the engine will start. 2) Spray your brake cleaner on the outside of the tubes one at a time. Eventually you will find the site of the leak as the car will surge in revs as it it sucks in the brake cleaner and ignites it. Finally check around the injector seats and also where the inlet manifold makes union with the cylinder head

Well, the secondary air intake malfunction has to do with the air pump I read. Water can get into the air pump in some models and the pump would have to be replaced. However, I have heard that some OEM pumps have a short hose that accumulates water sometimes, causing a recurring problem.

Also, there is sometimes a 20 or 30 Amps fuse that controls the pump.

However, I think, that even though they cleared the code at Advance Auto Parts, and it wouldn't start, that the no start condition is not related to the air pump problem, since the air pump problem is more of an emission control to reduce pollution and doesn't feed directly into the combusiton chamber.

I think that you should check the fuel injectors to make sure that they are working. They should click when under pressure with fuel, and allow fuel to enter the combustion chamber.

Maybe check the fuse for the FI or fuel injectors.

It seems like your vehicle is getting spark since it runs briefly when you spray brake cleaner into the air intake.

spray starter fluid into air intake and try to start it if it starts for a bit and dies you still arent getting fuel. Try changing fuel filter for starters. There are lots of things to check.

Try the WD-40 spray in intake of throttle body with air intake hose off, to see if it will kick at all, even for a second or 2. We need to confirm whether your losing fuel or spark when it dies. Ignition & fuel, can cause similar symptoms in this case.