My 328p-AU running on 12MHz can upload sketch only one time after burn bootloader (I use CH340g as USB uart to upload scketch).
1. With the new bootloader I am not able to upload any code the the MCU via serial interface. The chosen baud rate is set to 57600, which seems to be the cause of the issue. Changing it to 115200 in your boards.txt solves this.
If you have to increase the baud rate with a factor of two in order to communicate with it, it's likely that the chip is running twice as fast as it should.
2. My program generates some output via serial port. In my code the baud rate is set to 9600, but the output is garbled. If I change my terminal to 19200 baud, the output is OK. If I change my code to 4800 baud, then my terminal has to be set to 9600.
Same answer as 1. Are you 100% sure the crystal isn't running at 16 MHz? Because symptoms like this indicate that it is.
3. There is a difference in the low fuse values for the 168P. The original setting is 0xff(ext. crystal 8MHz). With MiniCore (168P, 8Mhz) it is changed to 0xf7 (ext. full-swing crystal). Changing it manually via ISP seems to have no effect either.
The full swing option consumes a little more power, but is able to drive "heaver" loads, and makes the chip less sensitive to interference on the XTAL lines. Full swing is the way to go for most applications that uses an external oscilator.