Two related questions:
I know on research that upgrading a magic item by the book costs the new item's cost minus the old's, in a nutshell. Would this also apply to creating a specific, named magic item if I knew it was constructed from a standard one, and provided that?
Generally speaking, if it's a craftable item (such as a
Luck Blade), and you have a suitable base item (such as a +2 Shortsword), then yes, the
Adding New Abilities clause lets you upgrade the base item to the improved item for the price difference (and crafting time). If you're upgrading it yourself, that's an 8,310 gp market for the +2 Short Sword, and a 22,060 gp market value for a Luck Blade (assuming 0 wishes...), so you pay half the difference in materials (13,750 gp is the difference, so 6,875 gp materials cost), one twenty-fifth the difference in XP (550 xp), and you take a number of eight-hour days equal to the market price difference over 1,000 (round up, so 14 days). If you're hiring someone else to do it, you pay the market price difference (13,750 gp) and wait the 14 days.
What if the enhancement bonus was not the same as the one specified?
Well, this depends somewhat on what you mean.
Taking the example of the Luck Blade again...
If you use a +1 Short Sword instead of a +2 Short Sword, no problem - you're just using a different base cost item when you're upgrading, so the math comes out a little different.
If you use a +3 Short Sword instead of a +2 Short Sword, you've got the issue that you're in "rules not explicit" territory - as far as I know, there's no rules for downgrading an item - so the answer is "ask your DM", as there is no RAW answer (that I know of - at least, baring specific things, such as the Artificer's Retain Essence class feature).
Secondarily, on the cost of creating an item with a special material: How would the cost be affected if I provided all the material necessary? I would assume there are still costs applicable for labor.
Well, per the
Craft Skill, materials are 1/3 of the market price of the item - so if you have, the materials suitable for making an adamantine greatsword, hiring someone to do it for you at market price would still cost you a little over 2,000 gp (3,050 gp * 2/3 = 2,033.3333.... gp, specifically, but who cares, really?)... and you'll need to wait a rather long time for it to be completed, with the way Craft skills work. If you
hire a suitably skilled Wizard to cast
Fabricate for you, at minimum caster level, you just have to pay the 450 gp.