About ATTINY85 Development Board
Is it really programmable withut any TTL/ICSP/FTDI extention?


Only if you use an ATTiny chip that has the "Micronucleus" bootloader already installed.


  If so, is it possible to program other AVR chip (Attiny/Atmega) with simple codes directly via USB...?


Some can, but most need a bootloader installing. The ATMega32U4, for example, has a "DFU" mode, which is essentially a bootloader in ROM, that you can use directly over USB.  In general, though, the answer is "Not without installing an appropriate bootloader with some hardware programmer first".