I would like to have different sidewalls for different type of tyres. For example, formula FOX have options SLICK_R1, SLICK_R2, ... I would like to have one sidewall for SLICK_R1, different for SLICK_R2, and so on. My question is: Is it possible somehow?

Well, choose the type of tyre that you want and then put the sidewall what you want.

No, there is no way to do that nowadays, would be cool to do F1 tyres...

Maybe my question wasn't clear enough. As I know it is possible to change type of tyres during the race when you do the pitstop. So, if you change for example your SLICK_R1 to SLICK_R2, I would like that tyre sidewall would also change (during the race). This way I would know which type of tyres someone has on his car. As Whiskey found out it would be ideal for F1 races, like in real Formula 1 (red tyres -> soft, white tyres -> hard).

it isnt. Even if you somehow made a insim that overwrites the sidewall dds when you change tires, it wouldn't change, because afaik LFS loads the dds only once on racestart.