Is belief in Allah's exaltedness, His establishment upon the Throne, and His nearness to His servants—without delving into matters of the spatial proximity of creation—sound and not contradictory to the texts of the Book and the Sunnah?

What you mentioned about believing in the exaltedness (uluw) of Allah Almighty and not delving into it is a correct belief. One should not give in to intense whisperings (waswas) concerning matters of creed, but rather suffice with what has been revealed in the Book (Quran) and the Sunnah.