InfoPath Dev

I have created a form in InfoPath Designer 2013. The form is in SharePoint in my XML Forms Library. Then I created a document library. Now, when I'm in my document library and click on “New”, I want the form, which is in the XML library, to be in the suggestions. I want the user to be able to save the form in the document library. In addition, the entries in the form should be generated in a SharePoint list.

You should probably publish the form to the library you created - under File > Publish > SharePoint while in design mode. Then you need to set up a submit connection in the form so that the user can submit the form to the document library. There is no out of the box way to also save the data to a list. You'll need to use code, CAML or qRules for that - if you are on SharePoint Online, none of those approaches will work, though there might be a way to do it with MS Flow.

Why does SharePoint Online make all that not work? I thought it was 100% compatible.

SharePoint Online no longer supports InfoPath Forms with code. It also doesn't support calling certain SharePoint web services - like the UpdateListItems method of the lists service - so CAML won't work there either. FormsViewer (a product built by the company I work for) can do this - more info here: https://www.qdabra.com/en/products/formsviewer.aspx