Since the real issue here is inflammation I will focus on the comment from the Wikipedia link you posted:
He also shows that arachidonic acid (AA, 20:4n-6) is an essential Omega-6 (1-6) polyunsaturated fatty acid that is abundant in skeletal muscle membrane phospholipids (figure 2). It is also the body's principal building block for the production of prostaglandins, which are known to have various physiological roles including a close involvement in inflammation.
Inflammatory prostaglandins do not have a close role in inflammation, they cause inflammation. When you injure an area of the body such as sprain your wrist prostaglandins dilate the blood vessels to improve blood flow to the area. But the overdilation of the blood vessels make them permeable and they leak fluids in to the surrounding tissues resulting in the redness, heat and swelling of inflammation. Bradykinin and white blood cells will also play roles in pain and infection fighting. So yes, the arachidonic acid does promote inflammation in part by increasing inflammatory prostaglandins. It also increases levels of inflammatory leukotrienes.
As far as the benefits of arachidonic acid, yes we do need SMALL amounts of arachidonic acid. This is why the body synthesizes what it needs from linoleic acid. But too much arachidonic acid clearly promotes inflammation, which among other things can lead to conditions such as heart disease. It is the same principle as water is essential for so many things in the body but too much can be harmful or deadly.
In addition high intake of arachidonic acid can lower levels of beneficial omega 3 fatty acids. For example:
In fact the omega 3s help to control inflammation in large part by interfering with arachidonic acid metabolism. So the omega 3s and arachidonic acid are antagonistic to each other. High levels of one will interfere with the other.