For those specific feats, technically a strike can be melee or ranged, so both feats work at range with this weapon.
With regards to the question in your title; 'Can a Halfling Sling Staff double as a Staff?'
Rules as Written (RAW), no you can't; you'd use it to make a ranged strike in melee, potentially causing an attack of opportunity, if a creature has that ability, and is also in range. Or possibly a GM might allow you to consider it an improvised weapon which would allow you to make melee attacks with it, without provoking, but at a -2 item penalty -- however that would raise the question of if you could apply the weapons runes or not, and RAW, I suspect not.
The only slight exception is that the feats that are part of the Staff Acrobat Archetype can refer to it as a Staff: '(referred to in this archetype as “your staff”)'. To me that is pretty clear in that it doesn't make it a staff -- it doesn't also let you treat a Spear as a staff.
As a GM, I would probably stick to the rules as written in this case, simply because getting to use those feats at range seems pretty good.