1. Should I leave unused pins unconnected or should I connect them to ground? Does grounding the pins have any positive effect?
No. Do not leave them floating. Strap to ground to prevent any possible EMI noise. Just good practice.
2. Should I leave unused pins unconnected, where the corresponding ports are internally used i.e. for USB, or should I connect them also to ground?
Ground for the same reason as noted above.
3. Can I pass signals (i.e. master clock) over unused pins, where the corresponding ports are internally used for USB, to avoid vias? Is using vias the better way to route lanes?
No. Use of vias is ok. Follow guidelines for USB High Speed PCB layout with impedance controlled traces. The PCB design must be impedance controlled during production and confirmed to be 90 ohms differential with a TDR test tool. The vendor can do this on demand at no cost or nominal fee (we have seen some offshore suppliers charge $10 USD for this service and will supply a printed report on how the traces are in compliance).
Best if you can share the schematic and PCB layout before submitting to a PCB shop for a quick review of the details.
Will you be producing the bare PCBs locally or offshore? The quality and capabilities vary a great deal offshore. We have used jlcpcb.com for a mix of simple designs but they cannot support complex layouts like HDI (via in pad / blind and buried vias) or small trace / small space. If possible, try to shoot for 5mil trace / space in your design. Use of 4L is a must for USB 2.0 HS designs and to reduce EMI.
Be sure to have in-rush current protection on the USB VBUS line and if possible, EMI inductors on the D+ / D- lines and a nice ESD suppressor - there are simple and low cost USB load switches that can work here. We use them all of the time on our USB products with success.