The #5 pins are GND pins on USB A. Do I need to beef up the GND traces connecting them? I am removing the vias now.
Beefier and shorter. They're the return for the current flowing through the device and need to be at least as wide as the traces on the #1 pins. What you did for the #6 shield pins, you should be doing for the #5 pins instead. The shield pins are a much lesser concern.
I am using a 2mm(78.74mils) trace for the source leads on MOSFETs, aren't they enough?
Looking at the MOSFET connected to VBUSA1 at the left, I see the drain trace is wide, which is good. The source trace connecting the two then heading east, which I presume is the big VOUTP bus, is still tiny.
I removed the traces on VOUT caps, instead I used a zone fill and punched vias directly on them. Do I need to follow the same for GND vias in all caps?
It would be wise to connect all the power caps solidly to ground, as you have done for the four BAT-GND caps by the inductor.
But their recommendation say to punch vias on free space to strengthen the connection between the layers and to help in heat dissipation. By routing together, Do you mean reducing the clearance between adjacent traces and routing them as close as possible?
Vias will help thermally just by being some metal, but would only help electrically if there's a ground fill on each layer to connect to, perhaps one that I'm not seeing because you turned it off for clarity for posting? (Aside, did you try setting the transparency of the copper layers to about 60%? I find that really helps me get a better idea of what's happening while laying out.)
Not necessarily reducing clearance, in the design rule sense, but yes, I suggest you run them closer together to keep the ground fill "traces" fat.
Can you name the caps?
C9,C7,C11, I think, where the concern is that trace looping nearly to the right edge of the board and dividing the ground plane between as it comes back toward the chip and passes between two caps. The top silk got a bit muddy in the JPEG compression, and to be completely honest I didn't see it at first, sorry.
