How to make an Arduino (ATMEGA16U4) like, software USB to serial on Atmega8u?
first question.

My assumption is that LUFA library helps to add software support to AVR which Don't have this kind of capability.

From the LUFA documentation, first sentence:

LUFA is an open-source USB library for the USB-enabled AVR microcontrollers,

So, this is only working on the chips that have USB hardware. Having some hardware doesn't mean you can do something useful with it.
Just like having a motor doesn't mean you can drive safely.
second question.

Could I use an AVR (one of those cheap types like atmega8a) as an intermediate device (to put between USB port and main Atmega328p)

Sure, but then you might as well use any USB-Serial bridge ICs that exist and get away cheaper, and more importantly, without debugging your own USB/serial stack (probably without the appropriate tools for that).
FTDI aren't the only people who build such ICs, just the most expensive ones. There are | other | vendors (and many more).
But really, come on. You're making your project here much harder than it needs to be. You want a microcontroller that can talk to USB. So buy a microcontroller that can, and comes with code examples, and has enough compute power to still do your main task while communicating over USB.
In other words, ditch ATMega8/16, and use some cheap 32 bit microcontroller that has USB and, best case, has FreeRTOS or mbed support with full USB support.