If the male OC Lannister was a proud and ambitious type, he would likely interact with Cersei with a mix of respect and rivalry. He'd try to gain her favor while also making sure she didn't see him as a threat. With Tyrion, he might initially look down on him but later come to respect his intelligence. And when it comes to Jon Snow, he could start off as dismissive but as he realizes Jon's leadership skills, he might try to form an alliance.
A more benevolent male OC Lannister would interact very differently. With Daenerys, he might be intrigued by her quest for the Iron Throne and try to broker a peace deal between her and the Lannisters. When it comes to Ned Stark, he could have a begrudging respect for Ned's sense of honor. And with Arya, he might be a bit fearful as he hears of her skills as an assassin, but also try to find a way to use her skills for his own ends, perhaps by offering her a pardon in exchange for her services.
Imagine a male OC Lannister who is a bit of a maverick. With Jaime, he would constantly challenge Jaime's loyalty to the family, asking him if his love for Cersei is clouding his judgment. When interacting with Sansa, he would see her as a valuable pawn in the game of thrones. He might try to marry her off to a Lannister ally to strengthen their position. And with Theon Greyjoy, he would view him as a weakling at first but later realize Theon's potential for redemption and try to use that to his advantage in the political turmoil of Westeros.