Where did the Prophet, peace and blessings be upon him, clip his nails and use the siwak? Is it considered distasteful to carry a siwak in one's pocket and use it on the road, during conversation, or while performing prayer? What is the ruling on this also concerning nails?

The Sunnah of the Prophet, peace and blessings be upon him, must be venerated, and objections to it based on corrupt opinions and intellects must be rejected. The siwak (tooth-stick) is a purificator for the mouth and pleasing to the Lord. The Prophet, peace and blessings be upon him, used the siwak upon entering his home, upon waking up, and he commanded its use for ablution and prayer. As for trimming nails, it is among the characteristics of the fitra (innate disposition), and there is no text specifying the location where the Prophet, peace and blessings be upon him, used to do it. The apparent understanding is that he used to do it in his home.