Technically, "your television set's big" should be considered correct as well. After all, it's okay to abbreviate the "is" in, e.g., "it's big."
Technically you can only abbreviate "is" with pronouns, but the colloquial usage is as you've written.
Just for practicing. If i want to say that the "television's screen" is big i would have to decline television to genitive?
"Twój(ego?) telewisory ekran jest duże"?
Or this phrase would have to be build differently?
Yeah, rather differently.
You start with the screen (the 'owned' thing), and then you have 'your television' (the 'owner'). "Ekran twojego telewizora". And the right form of the adjective is "duży", as "ekran" is masculine. "Ekran twojego telewizora jest duży".