BAB would be difficult as they don't have physical stats. This is probably why they typically are assigned buff/utility spells or attack spells that don't require such rolls (area of effect spells and the like). An intelligent item is already an big help in action economy, giving the character two actions in a round when it comes into play. Also despite the fact it says to treat the item like a construct, it already breaks away from that "type" in many ways, not independently mobile, no physical stats, doesn't have the HD(hit points)/saves/etc (as those are determined as per being a magical item).
Telekinetic Projectile is an orison so they probably assigned the BAB to keep it less effective than the typical "use CL modified by caster stat" that most other spells employee as it is usable at will by a "typical" caster. As it requires a BAB and a magical item doesn't actually have HD the spell basically becomes unusable to it, you would have to find a higher level spell that works with the stats it has (not what you wanted to hear I'm sure).
As it is written it would probably come out to 0 BAB (it doesn't have one no matter how you chop it, even as a construct it still has 0hd) modified by an effectively 0 dexterity (ranged attack) of -6 [0 Dex is unable to move but not unconscious]. Pretty underwhelming.
What I would probably do is assign it a 0 BAB and modify it with the highest mental ability score. You took the cheap way to get an extra attack and it shouldn't be very powerful essentially.
Sift would be untrained (nothing stated gives a magic item a class skill, and constructs never get them) and modified by the highest mental stat, yes.
Actually after sitting down and reading (instead of skimming earlier), skills would be based off the appropriate mental attributes. Intelligent magic items are unable to have anything besides Int/Wis/Cha skills unless they have a way to move as per the note on the ability table for magic items. So perception would be the total of skill ranks and modifier as an untrained check.