Hi dibbkd,
I’m using a 4000mah 3 cell LiPo in my F450 and it is perfect for it, the smaller barttey is necessary for the F330 to fit between frame plates.
I like to mount the battery between the frame plates because it keeps the CG and roll center where you want them.
That said, you actually need to get a 4000 mah battery that will fit between the frame plates and they come in a lot of different shapes and not all of them will fit.
I used a Turnigy Nano but I don’t recall which one specifically.
I left it in its as supplied protective bubble wrap and secured it with a velcro strap up through the bottom.
As for 4 cell, I would sauy outright for this copter no. These motors handle the lower voltage and higher current very well and will work very satisfactorily with up to 11x4.7 Carbon Filled Gem Fan Props,
For the 4S batteries you have to go to smaller props which are less efficient and will be noisier and yield shorter flight times for the same total battery power.
A 4000 mah 4S would definitely not fit between the frame plates.
You would need a 3000 mah 4S to equal the same power density as a 4000 mah 3S battery, but you would lose flight time because the smaller propellers (8" or 9") are much less efficient than the bigger slower 10" or 11" props.
If the motors or ESCs overheated from too high a current you could alleviate that by going 4S, but for these motors and ESCs it just isn’t a problem (they barely get warm) with the 3S regardless of weight or flying style.
If you are getting a Pixhawk it is important to get an RC system that supports PPM-Sum receivers.
There are a number of excellent solutions (and you should get an at least a 6 channel or better 7 or more channel transmitter).
The Turnigy 9X and 9XR transmitter from Hobbyking supports a plug in module from FRSky that will allow you to use PPM-Sum compatible FRSky receivers with it.
FR Sky sells a multiplatform (Delta 8) receiver that is compatible with Hitec and Futaba transmitters.
I use these with my Hitec Aurora 9 transmitter.
The excellent but hard to get currently FRSky Taranis transmitter is compatible with their PPM-Sum receivers.
And the popular Spektrum transmitter series is compatible by using only their “satellite” receiver although it is somewhat range limited.
There are others, but you need to ensure that the transmitters are compatible with appropriate PPM-SUM or SBus (serial) compatible receivers.
Modern 2.4 GHz, frequency agile RC systems actually have a considerable range generally well over a kilometer and often much further (line of sight).
If you go behind a building or mountain or something interferes between you and your transmitter all bets are off.
High frequency radios like a clear straight line path.
You sound like you are definitely heading in the right direction, I know when you are first getting into this the amount of information it is necessary to digest is pretty overwhelming - been there - done that.
Best of Luck, 
Gary
