Does this mean if I want the 3x a day use for this thing I divide the total cost by 15 (5 x 3 charges/day) making the over all cost 400 GP (6000 GP / 15 [3x/day]).
The cost of the item in question would result in the item pricing to 3,600 GP. (6,000 / (5/3)). I believe you may have accidentally multiplied where you should have divided, resulting in a larger divisor.
Also since this is made by 3rd Level caster, then all level dependent abilities of the spell used is = to the creators CL right?
Meaning this "X" of Blade Tutor's Spirit usable 3x/day, reduces attack penalty by 1 for only 3 minutes per use.
This is correct.
Is there way to calculate any extra cost to make the spell effect scale with the user?
So that the "X" of Blade Tutor's Spirit used by 15th level PC, the attack penalty is reduced by 4 (1 +1/5 Lvs) for a duration of 15 minute per use?
I am unaware of methods to create items using spells which automatically update to the current Caster Level of the original item creator - there are a smattering of classes and feats that grant a character a specific item that scales with them, and there are a handful of items and weapons that can become more powerful as its possessor gains power and/or takes appropriate "Relic" or "Legacy" feats.
Typically, this kind of problem is handled by the item creator making a new iteration of the desired magical item whenever he or she reaches a breaking point where the cost of duplicating a slightly better version of the same item is outweighed by its benefit.
However, Weapons of Legacy offers the following, which I present here without bias:
With the DM’s approval, a player might use a momentous event as the necessary catalyst to begin investing her character’s signature item with legacy abilities. She controls the process of imbuing the item with particular abilities as she advances in experience, within the larger context of the campaign’s plots and adventures.
While a Button of Blade Tutor's Spirit by itself might not seem thematically appropriate to be an "Item of Legacy," this at least offers you an avenue to explore with your DM regarding an item that can "scale" upward with your character. Of course, the book isn't correct for the system, and so if your table would not allow Weapons of Legacy (or other 3.5 only material) then the suggestion is moot.
Addendum: Apologies, Pathfinder in fact does have a method for creating scaling magical items. The full rules can be found in the Pathfinder SRD.
The table Value of Scaling Items gives us three categories: bauble, prize, and wonder. For the purpose of pricing this item, I assume bauble. A bauble that can hold abilities or spells which can scale to level 20 costs 44,000 GP Base Price on that table. Next, we add our spell. Referring to the table Spell Prices (One Use per Day), we see that a 1st level spell at 20th Caster Level would add 7,200 GP to that base price. (51, 200 GP) Now we follow the Crafting Scaling Items rule, which tells us to divide that final base price by 2. [51,200/2 = 25,600]. Thus, such a Bauble of Blade Tutor's Spirit, one usable once per day and scaling up to 20th level, would cost 25,600 GP to craft.