Why do the eyes of Utsava murti-s of thayar and Perumal appear as “closed”? Why is tilakam adorned to the utsava murti of Perumal (Examples – Namperumal, Varadar, Sarangapani, Parthasarathy) instead of Thiruman? In the processional palanquin of the utsava murti of Perumal and Thayar, what is the rationale behind the figure that looks like a bhootam?

A ritual called Netronmeelanam would have been performed for the eyes of Perumal and Thayar utsava murti-s. Based on this, it means their eyes are opened. If they appear closed, it is only so because they are structured. Mostly, Netronmeelanam implies that eyes are open.
Thiruman is an identification mark that proclaims that we are servants of Bhagavan. In this context, we cannot offer Thiruman to Perumal. He cannot become HIS servant. Hence, Thiruman is an alankaram, a decorative instrument for Him.
Have not seen Bhootam in the palanquin. Usually, have seen the head of a lion, presenting a vision that Perumal is seated in a Simhasanam. Simhamam is considered superior, and this is a mark that reveals the superiority.