Well if you can hear it then the system is charging so that eliminates the fuel pump, it's connections and the relays... It even eliminates the fuel cut off switch...
The hiss when releasing the fuel cap isn't always there, it means the evap system is working too but after attempts at cranking it usually builds up then dissipates via the evap canister pushing the fumes into the inlet manifold.
It definitely sounds like an electrical contact issue though.. Engines vibrate and mounts can have a bit play that pulls on cables....
If it was mine, I would start by removing the battery putting it on charge and while waiting, disconnecting each sensor and multiplug one at a time.. cleaning up the contacts, giving them a spray with contact cleaner and then refitting before moving to the next one..
Don't forget the fuel injection plugs, the big main engine loom cylinder type multiplug and the coil packs.
One last thing... The earth straps.. one attached to the gearbox in particular, remove inspect clean its ends and refit.
Move onto the relays , take them out one at a time and clean both male and female contact's.... You may stumble upon the problem half way through but keep at it and finish off cleaning and recurring, it will pay for itself in the long term with trouble free electrics for years to come
Before fitting the battery, disassemble the positive terminal fuse box, take a photo of what goes where if you aren't sure and clean it up. Make sure that the rear positive terminal has both rings top and bottom on the battery post before nipping up.
The battery negative, follow it, remove, clean up both ends and the bolt and refit.
Now try starting her up.
Now, apart from your time and the price of a can of electrical contact cleaner, that was all FREE TO DO.
If in the unlikely event that hasn't done the trick. Remove the front bank of plugs and check to see if they are wet with fuel which means you have definate fuel pressure but no spark which leads to coil packs failure.
When you have everything disconnected for access seal the rubber induction hose with river gloves and elastic bands and submerge it in a bucket to check for leaks... Anything leaking after a maf will cause really bad running and possible stalling. That alone though won't throw up a warning light... Thus all the checks and information above.....
Good luck, don't be disheartened, you are helping with prevention which is better than trying to find a cure in the future.