♐ Out of guilt for his failure to spare Nepeta from death, Equius wants to be the body guard of Karkat, or something to that effect. Karkat meant the most to Nepeta so he feels as though this will somehow alleviate his guilt.
♐ Another reason for his desire to do good for Karkat lies in his hallucinations. Equius often hears Nepeta speaking to him and touching him. He gets caught up in these fantasies and sometimes forgets that Nepeta is not really there. 
♐  Flushed feelings would be extremely difficult to come by in Equius. He is mostly full of anguish now so it is much easier for him. However he IS capable of red feelings. He also feels that engaging in a red quadrant with another troll would be a betrayal to Nepeta. As Nepeta was the most important troll to Equius, he will not allow for anytroll to replace her as most important to him.