lincoln logs
No, Abraham Lincoln did not invent cornbread. Cornbread was a staple food throughout the Americas centuries before Lincoln's birth.
Abraham Lincoln did not invent any of the items you refer to in the question. He was the 16th president of the USA, not an inventor.
Yes. Abe used to live in a log cabin.
No, he did not.
That would be Abraham Lincoln, and the toy Lincoln Logs.
The toy is Lincoln Logs, named after Abraham Lincoln.
No, Abraham Lincoln did not invent peanut butter.
Yes
Although they carry Lincoln's name, Abraham Lincoln was not involved. This popular toy was invented in about 1916, and undoubtedly carried Lincoln's name because the former president lived in a log cabin (and we are told he built it himself), so the toymaker figured this clever name would help everyone to remember the toy when they bought it. Lincoln logs allowed children to build things the way Lincoln did, making replicas of log cabins and other famous residences.
Lincoln Logs was created in 1918.
Lincoln Logs, named after Abraham Lincoln (because he grew up in a log house).