I can't explain many things since I don't want people to link the non erotic HTA to these games so I will keep it minimal.
About Arch's role, Merusa turns evil because of his lust for power. Merusa curses his lover to become cumslut for everyone (you know how Momoko's husband treat her) and imprisons him. He makes the ring hoping to use it to become powerful enough to escape his captivity. When Lust comes to him for help he asks for her power to fuel his ring which is why ring is focused on sex.
In current time, he asks Masuo to corrupt ladies that would never betray their husband represented by hearts icon on his throne. This will make him powerful enough to escape his prison. You will see a pink heart already glowing because Masuo has corrupted Momoko's mind.
Nadeko who serves Merusa is also affected by this corruption becomes twisted and eventually wishes to be a God. She meets Arch and he sweet-talked her into taking over Merusa and gives her the ring. She fails to take over Merusa and gets imprisoned in Merusa's staff. Hailey's goal is finding that staff.
Ryoko is not his student. She is part of Mystery's group.
Reita's job is to eliminate champions of other Gods. He eliminates all males and copulates with females since he knows he can influence them eventually. Lady Mystery has such champions she can contact when the time is right. Since Ryoko is never born in that universe she is out of the question but others are Himawari, Sugar, Izumi, Rumia and Elaine.
Mikasa's past is played in HTA 2. Mikasa and her brother becomes Merusa's champion after she sees Himawari defeating her guardian. Nadeko hands over the ring she got from Arch to Mikasa's brother hoping he would becomes strong and eventually she does.
She and brother fights Himawari where he dies. Mikasa wanted to avenge her brother but seeing Masuo coming to Himawari aid, seeing him runs away in disbelief (she met Masuo way before and learns he is decendant of Karasuyama family). She wears ring knowing it will make her strong and will become like her brother as long as she doesn't engage in sex.
Mikata and Mikano are versions of Mikasa from Ravaging beast and Street Hound.