He claims that she is his. In episode five, when he looked up at Ritsuka's crying face, he got flustered and started blushing. He presumably may like her. When he was battling Lindo he claimed that he would destroy anyone who got in his way, but when he and Lindo prepared to kill each other, and Ritsuka cast herself between them, he pulled back in horror. He is then seen in the third library reminiscing on how sweet Ritsuka smelled and the time he spent with her.
It is further confirmed in episode 8 when he believes that she doesn't know anything about the grimoire, his interest in her only increases. He is shown blushing when he asks Ritsuka to be his partner at the dance. He also gets jealous when she and Rem dance together.
In episode 9, he is shown throughout the episode looking for Ritsuka once he learns that she is in fact the grimoire. He fights the exorcists, along with Urie and Shiki, when they threaten to kill Ritsuka. In episode 10, Mage gets visibly jealous when Shiki and Urie say they love her. He claims that she is his, and that he wants her laugh, teary face—all of her! He later becomes protective when he agrees with Urie that they can't let Rem near her as long as he denies his feelings.
In episode 11, when everyone goes to save her from the vampires, Mage says that he loves her and "means it, for real," asking her to believe him and not say "it's not like him at all." She is touched by his confession. In episode 12, he fights to protect her, and while he is saddened by her and Rem's mutual love confession in the end, he is accepting of it. However, before he leaves, he promises to return and make Ritsuka his one day, signifying that his love for her is eternal, and that she will always be the only girl he desires.
Mage has a begrudging respect for Rem but does not enjoy being considered one of his pawns. In episode 5, he tells Rem that he will not be one of his chess pieces anymore, and tells him to go "play his little games by himself."
He is shown to be jealous of Rem in episode 8, whenever he and Ritsuka dance together. He later demands that Rem confess that he loves her too. When he refuses, Mage is shown agreeing that they can't let him have her as long as he lies. They can't trust that he won't hurt her. He later reconciles and teams up with him to save Ritsuka.
Mage and Urie have something of a rivalry between them, being polar opposites. They literally go head-to-head in episode 2 over their quirks. (i.e. Mage's manliness and Urie's flirtatious nature) Then, in episode 8, when both of them ask Ritsuka to be their partner at the dance, they literally butt heads over it once again. They get jealous of each other when they both declare their love for her. However, they also both agree with the other when they say that until Rem tells the truth about his feelings, then they can't trust him with Ritsuka. The two band together with the others to save Ritsuka.
Not much is known about Mage and Shiki's relationship. They seem to be on friendly yet annoyed terms, with Shiki often popping up out of nowhere to tease Mage. In episode 5, when Mage is reminiscing on his time with Ritsuka—smelling the place on his shirt where he was holding her—Shiki emerges from the shadows making him jump. He then teases Mage on how red his face is after thinking of Ritsuka. Mage then angrily denies the obvious, telling Shiki that if he brings it up again he will murder him. Shiki brushes this off, saying that he wished that he would kill him, but would have to pass at the moment. This only served to annoy Mage even more.
Mage considers Shiki beneath him, because of his fallen angel status, and never fails to pass up an opportunity to mock whim about it. He often threatens ShikI, and dislikes it whenever he tries to get ahead of him. However, he did team up with him and Urie to fight the exorcists whenever they threatened to kill Ritsuka. In episode 10, he threatens Shiki when he asks Rem to let him have Ritsuka. However, they later agree with each other when Urie says that they can't let Rem have Ritsuka without knowing his true feelings. They all later band together to save Ritsuka from the vampires.