I am making my own circuit and I am using ATtiny10-TS8R (I couldn't buy any another type for now), but I found (6) in the datasheet on the part I want to use:
As noted in point (6), it says "Not recommended for new designs".
I used it already and the circuit worked very well, but I am willing to go to mass production, so, is it safe to use that one (TS8R) or may my circuit face problems in the future?
Not recommended for new designs means just that - they want to remove that SKU and are warning you that it will happen soon. If you are producing a large volume (often not actually that big, >10K), they might continue manufacturing for you on a contract basis.
This warning suggests you should move to a compatible or newer microcontroller for a design you want to start producing now and into the future. In this case, it would be the "ATTINY10-TSFR" if you want to have the same temperature range. If you can secure enough stock of the obsolete part for your purposes, then go for it, but be aware they might not be available in the future.