Thanks for pointing me in that direction. I had searched the forums for an answer all last week, but didn't post my question until this week. In the intervening time, that other thread answered my main question.
I have changed the MISO, MOSI, etc. to the correct pins for the Mega2560. This improved the situation to a degree. At least now the error messages are consistent.
When I try to burn the bootloader to the ATTiny45 @ 8 MHz, I get the error stk500_initialize(): programmer not in sync, resp=0x15. The same error occurs trying to burn the bootloader as an ATTiny45 @ 1 MHz. When I try to upload a sketch with the board set to the ATTiny at either 1 MHz or 8 MHz, I again get the same error.
I have tried changing the baud rate from 19200 to 9600, but that hasn't helped.
I am using the ArduinoISP for the programmer, and I have a 10uF capacitor between +5V RESET and GND to stop the Arduino from resetting.
Any thoughts on what might be causing this?