In By the Waters of Babylon, where did John find the painting of flowers and the word "Lincoln"?

I do not think that it is possible to answer either of these questions in any specific way.  There is no evidence from the story to give us a definite answer.

The painting of the flowers is clearly some sort of pontillist painting like those of Georges Seurat--these are the kind that are made of dots of color that look like dots up close but blur together into pictures when seen from farther away.  But there is no way to say what particular painting this is.

Lincoln's name only comes up at the end of the story.  John mentions the name as one of the "gods" that built the city, but he does not say anything about where he saw the name.

Where did John see the word "Lincoln" in "By the Waters of Babylon"?

I've read through the entire story very closely, and don't see any place where he sees the word Lincoln. However, he does come across a ruined statue with the letters "ASHING" under it.  He describes the statue as a tall man, with funny clothes on, and his hair tied in the back like a woman's.  If we take those clues, and try to piece them together with the letters on the statue, we can conclude that the statue was probably one of George Washington, the first president of the United States.  In our society, we revere this great man, and have statues of him all over the place.  That one of them survived, partially at least, is interesting.  So, even though Lincoln wasn't mentioned in the story (to my knowledge), George Washington was.  They were both presidents, and men whose presence is often represented in art and statues, so that is probably where the confusion came from.  I hope that those thoughts helped; good luck!