Alright, it took about 2 days, but I can sit down with my computer now...
Attune Item: This one is potentially an oversight, but on closer reading I don't think it is. The first part, choosing the item and performing the proper skill test to perform the ritual is pretty clearly laid out. It specifically calls out Instruments and lists the proper skill and everything. The part that you called out as being contradictory is only partially so. It just so happens that attuning an instrument doesn't inherently improve your ability to perform (showmanship). It might allow the adept to perform amazing feats of technical skill with the instrument, but not charisma-based performance. At least, that's the way I read it which follows RAW. Thematically, if a GM wants to override with the exception that Performance can also be affected, that would also be reasonable. It is very possible that the writer of this ritual overlooked or forgot that Performance is not paired with a physical attribute or the passage was cut to save words (it is quite lengthy, even as is).
Personally, I would rule that an adept that has attuned an instrument can make Performance + Agility tests while benefiting from the limit increase. But there are some further benefits for attuning an item, so I wouldn't call it a contradiction. Not all items benefit the same way from being attuned.
Increase Gear Limit: Already discussed.
Rectitude: Ok, it took a little bit of looking, but I figured out the confusion here. The Imbue Item ritual, which is what allows you to apply those traits from page 132 is not limited to items that are attuned using the Attune Item (Adept) ritual. Anything magical which can be bonded by an awakened individual is referred to as being "attuned." The Adept ritual (Attune Item) simply allows an adept to bestow a magical-like status to a mundane object and thus attune it. Not all of the Imbued Traits apply to these adept attuned items, the first one for example (Alliance) explicitly doesn't.
In the case of Rectitude, the wording is somewhat poor (true), but the phrasing clearly indicates that it is an imbued trait that can only be applied to Qi or Weapon Foci. Since the Imbue Item ritual can be used on foci as well as adept attuned items, this is totally allowed.