The random alignment of the tetrads along the equatorial plate; occurs during metaphase I?

Yes, the random alignment of the tetrads along with the equatorial plate occurs during metaphase I.

Explanation:

After the crossing-over in prophase I, and during metaphase I, fibers of the spindle apparatus capture chromosomes and take them toward the center of the cell, to the metaphase plate. The homologous pairs of chromosomes get aligned in the metaphase plate for their posterior separation in anaphase I.

Each chromosome of the pair joins with a microtubule of only one of the spindle poles, so the two equivalent chromosomes join the fibers of opposites poles.

When they align in the center of the cell, their orientation is random.