[Feature Request]: UX: Make the `Run` panel dockable to side bar

### Is there an existing issue for this?

- [x] I have searched the existing issues and checked the recent builds/commits

### What would your feature do ?

Currently, the panel with `Run` button (I'm not sure how it's called) can only be in 2 places:
- docked in the main menu
- floating

The side bar has a plenty of unused space - why not allow docking it there, too?

### Proposed workflow

1. A user can drag-n-drop the panel on the side bar.
2. It gets docked there, becoming arranged vertically.
3. Panel-detaching handle acts as a separator from the main set of icons.
4. Run button can get rid of text and keep only an icon.
5. The triple-widget combo of run button + a dropdown-like triangle after it + a text field for number of batches remain grouped, but get arranged vertically, too.

### Additional information

_No response_

┆Issue is synchronized with this [Notion page](https://www.notion.so/Issue-4344-Feature-Request-UX-Make-the-Run-panel-dockable-to-side-bar-2266d73d3650816d99f5f2a1a9da6c88) by [Unito](https://www.unito.io)