The main help for you is following all steps from the tutorial How to Make Hotel Booking Calendar with dhtmlxScheduler to understand how it was created.
Filtering sections described in Step 1 – Filtering timeline rows (rooms) in the tutorial.
Show paid, i.e. filtering events, described in the documentation here.
To print the document, you should use the export to PDF/PNG and then print the generated file.
To add an icon to header of the sections, please find Step 3 – Adding a header to the lefthand column of the timeline in the tutorial. You need to put html element for the icon into the
headerHTML varible and specify CSS for the icon.
Lines can be customized by redefining CSS.
Select controls in the lightbox described here.
Check-in date/time can be creates as a Custom Lightbox Control.
Price, Paid, Notes can be added as Textarea controls.
There is the lightbox Time control for time. The “Nights” field should be added additionaly, also using the Textarea, for example.