Short answer Lex is running Luthor Corp out of National City. To the general public he's a hero who won the Nobel Peace prize. He's having fun messing with Kara and Lena but recently has been clued into Leviathan and believes the organization is a threat to him. Clark hasn't been mentioned at all but we've only had two post CoIE episodes and they were filmed back in November. Long answer how much if any of Jon Cyrer's Lex appears on Superman and Lois is entirely up to Cryer. He's always listed as a special guest of Supergirl. He's an expensive actor for a CW show yet he doesn't need the money. He's likely taking a pay cut because he's enjoying himself. He asked to be in CoIE and from interviews he's having a ball with the part. Part of that fun however is working with Melissa Benoist and Katie McGrath. He's an actor who enjoys working with people at or near his level especially if they're "giving" to the other actors in the scene. Both Melissa and Katie are known for being "giving" to their scene partners. If Cryer can't have the same fun on Superman and Lois he's got no reason to be a significant part of the show.