Can I use mbed for STM32F103(C8T6)?
Yes, you can. 
You can create a new project based on NUCLEO F103RB and then replace precompiled library with sources. Then in sources you can redefine pin mappings to fit your specific board.
Please note, that I haven't done this for STM MCUs, but ~2 years ago I got it working for my nRF51 custom board.

The idea is, that MCU stays the same with all the internal peripherals, you just have pins wired slightly differently. Additional changes might be required if your board has different clock source.