I am combining the trivial 'leave Shell input SyntaxWarnings alone (instead of making them SyntaxErrors)' part of #34857 with this issue, 'print Shell input (and the very rare internal IDLE Warnings occurring after Shell exists) in Shell, and changing the title accordingly.

The PR is a WIP with at least two bugs. Help wanted.
1. the triple output.  (Before the patch, when I started IDLE from command prompt, I only saw one.
2. Warnings are inserted before the text generating the warning.  I interpret this as intercepted \n causing compilation and printing of the warning before the \n is inserted at the end of the line and the iomark moved to the beginning of the next.  I don't know what moves the iomark.