Yes and yes.
Despite a persistant belief to the contrary, an employer may provide any information that is true, that they they honestly believe is true, or that represents their honest and supportable opinion. The myth that an employer is limited to dates of employment, salary and job title (sometimes including rehire status) is just that - a myth.
An employer may deny you a job based on your performance, including behavior, in a previous job. That includes "issues" with management.

