Things are a bit more complicated than that the character will simply enter public domain. First off, it'd be like the Superman (characterization, power set, etc.) in Action Comics #1 that is in public domain. Later added elements (additional powers, characters (ex. Zod, Kon-El, Maggie Sawyer, Doomsday, Ron Troupe, Supergirl), and stuff like the Fortress of Solitude) would still fall under copyright.
When someone is using the public domain version of Lois, they'd have to line her up with her Action Comics #1 characterization. If they write her like the Donnerverse, Lois & Clark, Smallville, or DCEU Lois Lanes (possibly even stuff like the John Byrne revamped Lois Lane from the 1980s), they'd be in violation of copyright law.
The "S"-shield has been redesigned numerous times over the decades. If you use the wrong one (ex. the Kingdom Come shield, introduced/created in the 1990s), you'd be in violation of copyright law.
So, no, DC would have nothing to fear. As anyone else using Superman would be unable to use any of the stuff from the 1950s, 1960s, 1970s, 1980s, and so on (which would include the majority of his rogues gallery). If someone writes a story, where Superman goes up against Doomsday (introduced in the 1990s), they'd be in violation of copyright law (and DC could just sue this person).