Should we do sankalpam mentioning the tithi (date), vAra (day), and nakshatram (star) during Salagrama Tiruvaradhanam? When should pratha-sandhya be performed before Dhanur month aradhanam?

Yes, it is in practice to mention the day, date, and star during Salagrama Tiruvaradhanam.
Dhanur month aradhanam should be performed before sunrise after which Sandhyavandanam is done. If the sunrise has already occurred, Sandhyavandanam should be performed first followed by Dhanur month Tiruvaradhanam.