How to wrap a contact form with another block?
Have a look at either of the contributed modules below (quotes are from their project pages).
Contact Block

... provides contact forms in a block. It uses the contact forms that can be created using the Drupal 8 core Contact module.


Release info: only for D8.
Entityform block

Allows to load a specified entity form as a block.
Usage: Go to the Block layout admin page (admin/structure/block), press Place block in the desired region and select Entity form.

Release info: Either for D7, or D8 (but only an alfa release for D8 right now).