sin should normally be pronunced that way when it appears after a word ending in an r. We usually assimilate r + s over word borders too. (some dialects have special R:s and don't do this, but Standard Swedish does).
Yes, but the meaning is different: your translation would mean he writes some other guy's name, whereas "sin" means it is his own name that he writes. English is actually ambiguous in this case, so either could be a valid translation from the English.
Why han skriver sin ålder and hon skriver hennes alder. The second sentence should mean the woman writes some othern woman's age an d not her own age. But in English her is used both for own possession even for others possessions