Can I customize the contact relationship tab to show custom fields?
Which CiviCRM version are you using? If 4.4, then it looks as though this extension may do what you want:

Custom Contact Relationships Tab

If you're using a more recent CiviCRM version, it may be worth contacting the maintainer of that extension to see whether they are working on support for more recent versions.

There was some discussion of this feature on the old Civi forum in 2013:
Changing relationship fields on the relationship tab. But please continue the discussion here rather than there. :-)