I am coaching a 21 year old athlete with a very high arched foot. It seems his foot structure is causing all sorts of lower limb injuries: shin splints, PF, weak ankles and most recently achilles problems."
All of what you will read above is only relevant to each of those individuals.
As your athlete runs toward you, are their hips abducted?
Can their foot compensate to 90 degrees? Hows the dorsiflexion?
Do you attempt recovery-mechanic drills to wrap the feet beneath the athlete?
If the athlete is rotating heavily and quickly across the ball of the foot- then of course you will have gastroc tightness as a result of the movement.
A Pes Cavus Foot in most cases caused the forefoot to be slightly varus- the subtalor joing would be somewhat over "supinated" upon footstrike assuming they are indeed a forefoot striker. Of course there will be achilles problems if the athlete is loading the forefoot during the stance phase to propulsion phase (a quick unstable heavy rotation acrosst the ball of the foot).
The PF is likely just from the foot pronating- with a lack of adjustment from the posterior tib- something has to give. Therefore the dick littles above throwing around shoe advice are absolutely wrong. Try a shoe with a medial forefoot wedge. (NB 1223) Or just take the kid and get him orthotics with a medial forefoot wedge.
Also- pick up a book. Stay off the message boards for orthopod advice. It will prolong a resolution.
The half-assed advice tossed around by know-it-alls in regard to foot problems is "hilarious".