How does my mother apply for my father's pension?

Why at 86 is your Mom not receiving his pension? If he has passed, she needed to change his status at the time he passed. At that time she claimes survivor benefits. When he retired, Dad was asked if he wanted survivor benefits. If he said yes, he got less in pension. To get his benefits, it has to show her as a survivor. You need to call the benefits office at your fathers previous employer.

If Mom is low income and fits Medicaid criteria, you may be able to get help there. Or if Mom has money, she can pay you.

You will have to read the documents that pertain to the pension and call the company. They will tell you how to do it. These things differ.
I am not certain what you mean by "how do we go about getting help for me being her caregiver". What kind of help are you needing and wanting?