Can I transfer PDF files to a USB flash drive on a iPad 2 w/ OS 9.3.5?

No.

There are a few special storage devices for mobile devices. There are portable, bulit-in WiFi hotspot hard drives for mobile devices made by Western Digital and Seagate.

http://www.wdc.com/en/products/products.aspx?id=1330

http://www.wdc.com/en/products/products.aspx?id=1660

http://www.seagate.com/consumer/stream/

There are special mobile lightning connected and built in WiFi Hot spot, USB flash drives made by SanDisk (now owned by Western Digital)

https://www.sandisk.com/home/mobile-device-storage/ixpand

https://www.sandisk.com/home/mobile-device-storage/connect-wireless-stick

http://www.adamelements.com/iKlips/iklips.html

This listing maybe incomplete and somewhat outdated, now. So do an online search for similar external data storage devices for iPads

Kingston Electronics makes some data storage devices for iPads/iDevices, now, too!

Good Luck!