@CodingBadly, when you say "328 family bootloaders... do not support accessing the EEPROM", do you mean to highlight that as a current limitation of those bootloaders, and something to keep in mind if I plan to do anything with the bootloader code ?
Mostly. "...something to keep in mind if I plan to access the EEPROM on my Uno" is a more accurate
Does it mean that if a Duemilanove or Uno is used as the target (not as the "Arduino-as-ISP board"), directly connected to PC via Serial-over-USB, I can't directly write to EEPROM ?
Exactly.
1. Use Arduino-as-ISP (or even willing to invest in that nice Pololu USB ISP), to program the target ATtiny85, and also write/read information on it's EEPROM from the PC.
a. Is this possible ?
Yes.
b. Can I use avrdude command-line version using Arduino-as-ISP to read / write EEPROM ? Or I need a dedicated ISP ?
A dedicated ISP is not required (but is often handy

).
2. Somehow (not sure how), target Arduino board (Uno / Duemilanove)'s EEPROM, to be read / written from PC.
a. Is this possible ?
It is if you put an ISP in the middle. A second board running ArduinoISP will work.
The Arduinos have a six-pin ISP connector just for that sort of work or you can connect jumpers to the correct pins.
b. Can I use avrdude command-line version for this ?
Yes. The commands are almost identical. The only thing that changes is the kind of memory (Flash vs EEPROM vs Signature).
I guess not, because bootloader doesn't support it ?
That is only an issue if you try to access EEPROM on a board connect directly to the computer.
Essentially, the Arduino bootloaders are like a carry-along ISP. Wherever the processor goes, an ISP (in the form of a bootloader) goes with it. The bootloader in question is very basic; it lacks EEPROM support. Support could be added but most people do not need it and EEPROM support would make the bootloader considerably larger.
Hopefully, I could reset the fuse-bits (if set) that erase EEPROM when programming the flash, using avrdude commandline, and either a standalone ISP or Arduino-as-ISP.
You can. But I suggest first checking if it is an issue for the target board. All the new processors I've purchased have "erase EEPROM" set (the fuses have to be changed to "preserve EEPROM").
If you are using the Tiny Core and ever Burn Bootloader, "preserve EEPROM" is set.