How to display a contact form as a node block?
For non coding solution, you can go with

Create the form using Webform and utilizing Webform Blocks and Token


  Webform is the module for making forms and surveys in Drupal. After a submission customizable e-mails can be sent to administrators and/or submitters. Results can be exported into Excel or other spreadsheet applications. Webform also provides some basic statistical review and has an extensive API for expanding its features.



Form Block


  Enables the presentation of user registration, site wide contact, or
  node creation forms in blocks. This is particularly useful for
  including forms on panels.


Worth mentioning, just might not do for this question's specific use case.


Do it programmatically by creating a custom block and fetching the form, you can find an example here Problems to integrate a form into a block