How can I run Cuphead version 1.1? I know how to run the legacy version, but I don't know how to run the 1.1 version. And I have an additional question. The Livesplit timer stops, but it doesn't split automatically. And the timer doesn't stop after the Sally Stageply boss ends.

You just have to install it on steam.

1.1 and Legacy are accessed the same way, through your Steam Library, then Cuphead's Settings (to the left of the Info icon), then Properties, than Betas. Once here, select the option I have put an arrow next to in the following screenshot (this may look slightly different for you as the screenshot was taken on a Mac)

btw not fully sure what's going on with the Loadless timer. Would be of good use for you to ask abt this in the Discord.