Whenever the "name" field shows up (records, relationships, queries, exports, mailings, reports, etc.), it should include the suffix. I've had to recreate several records that were merged because a father and a son share a name, and live at the same address, and give us the same email address.
Two similar topics "View Contact Name suffix on Action records and on Query output results tab" (http://rediscovery.uservoice.com/forums/137015-raiser-s-edge-discovery-topics/suggestions/3792747-view-contact-name-suffix-on-action-records-and-on) and "More information displayed on soft credit tab" (http://rediscovery.uservoice.com/forums/137015-raiser-s-edge-discovery-topics/suggestions/5901980-more-information-displayed-on-soft-credit-tab) should be merged with this one.
You can set up/use an addressee/salutation format that includes the suffix and then choose to use that for your queries/exports/mailings/reports etc. I changed my default formats to what we wanted.
I completely agree with adding the suffix to the "Name" field. Of course we can add Addressee fields to a query, etc, but the problem is when you are looking at a list of constituents (query list) it does not have the option to show addressee, just name.