Does this mean that I need to be completely finished with kashering my
home, and have all chametz out by 12:12PM?
You need to be finished kashering by Noon, which will be about an hour after you have to be done with burning your Chametz.
You need to have all chametz sold or destroyed by 12:12PM.
If I were to encounter chametz that was not seen prior to 12:12PM,
could I still remove it after 12:12PM?
Chametz found after 12:12PM must be destroyed immediately. Preferably by fire. Do not eat it!
Also, if the kashering of an oven or dishes hadn't quite reached the
24-hour mark since the physical kashering, would the appliance or
dishes be considered treif and chametz by 12:12PM?
The 24-hour period is needed before kashering in boiling water. Kashering in boiling water after Noon on Erev Pessach, or during Pessach would require Rabbinic guidance, if it's at all possible.
Kashering by fire is allowed after Noon.
The preferred route is to make sure everything is kashered before Noon; thus stop using it before Noon the day before.
Dishes not kashered are Chametz and cannot be used during Pessach. They are not Treif, and can be used after Pessach without further ado if they were clean.
All answers based in the Kitzur Shulchan Aruch 116.