[RPG] Why would I buy Darkleaf Hide armor instead of a Mithril Chain Shirt

Quickly over viewing this are the answers I can give:

1. The Armoured Kilt has a maximum dexterity bonus of +6, I am unsure of what armour you are also wearing but your maximum dexterity bonus is the lowest of the two figures.
2. The Armoured Kilt has no armour check penalty so it should not affect your swimming or climb checks, regarding land speed Can Canbek's answer below has now covered it as well as clearing up any confusion on the matter regarding armour properties.

3. This should not have an adverse effect on the barbarians improved speed:

   Fast Movement (Ex)

   A barbarian's land speed is faster than the norm for her race by +10 feet. This benefit applies only when he is wearing no armor, light armor, or medium armor, and not carrying a heavy load. Apply this bonus before modifying the barbarian's speed because of any load carried or armor worn. This bonus stacks with any other bonuses to the barbarian's land speed.

   As described the Fast Movement (Ex) ability continues to work in medium armour.

You're misusing the guidelines.

First, let's take a look at what the Pathfinder Magic Item Gold Piece Values section has to say on this issue. Turns out, it is specifically addressed.

The correct way to price an item is by comparing its abilities to similar items (see Magic Item Gold Piece Values), and only if there are no similar items should you use the pricing formulas to determine an approximate price for the item. If you discover a loophole that allows an item to have an ability for a much lower price than is given for a comparable item, the GM should require using the price of the item, as that is the standard cost for such an effect. Most of these loopholes stem from trying to get unlimited uses per day of a spell effect from the "command word" or "use-activated or continuous" lines of Table: Estimating Magic Item Gold Piece Values.

Emphasis mine. It then goes on to take Mage Armor as an example of this:

Patrick's wizard wants to create bracers with a continuous mage armor ability, granting the wearer a +4 armor bonus to AC. The formula indicates this would cost 2,000 gp (spell level 1, caster level 1). Jessica reminds him that bracers of armor +4 are priced at 16,000 gp and Patrick's bracers should have that price as well. Patrick agrees, and because he only has 2,000 gp to spend, he decides to spend 1,000 gp of that to craft bracers of armor +1 using the standard bracer prices.

So yes, your Amulet of Mage Armor and Shield is a textbook example of the guidelines gone bad.

This is understandable. The guidelines are famously weak, often breaking with casual use, as you have encountered. This precise case, however, was anticipated and addressed.