Since skjorte has the feminine form skjorta the possessive pronoun is si, it points back to the gender of the object (the shirt). Skjorte can also get the masculine form skjorten (it's a dialectical preference whether to choose m or f) and the possessive would be sin
Sitt applies to neuter nouns.
I use rense about cleaning stuff without the use of water/fluids (or with minimal amount of fluids). And sometimes if it is an especially thorough sort of washing. I would normally be satisfied with washing my hair (vaske håret), but every now and then I'd want to cleanse it (rense håret). Yes, I just discovered that we would say rense for both cleanse and clean.
You wash your hands and clean your nails. Du vasker hendene og renser neglene.
A dry-cleaner is et renseri.