Ok, in the first pic, i think I see a undulated triggerfish. Get him out right away since they are extremely aggresive and it will kill everything in there eventualy. Some might be said of the clown triggerfish but there are at least 50/50. Most people keep undulated in a 55 gallon by themselves. Clown trigger will also need a bigger tank, around 150 gallons or so. Are you planning on upgrading? Even if you are, I would reccomend taking out quite a few fish. What are the dimensions of your 60 gallon? It looks like a 60 high. Take out all the tangs, the blues, yellow, and brown powder . The blue and the yellow need at least a 75 gallon stardard. The Powder Brown Tang needs to have some live rock in there and it also needs a bigger tank. . The foxface might be ok for now, but will need a bigger tank or at least a different dimension tank. The angelfish both need bigger tanks and live rock to graze upon, they are also not reef safe and will nip at your corals and clam. What kind of lighting is on your tank? There looks to be a clam in there, and they need high intensity lighting.
Heres what I would do: Return all fish(including damsels) except the foxface, maroon clown, and the smaller blue tang. Buy about 70 lbs of cured live rock and put into system. Buy a clean up crew of snails and hermits. Your tank is made for corals, IMO and small reef fish. Hope this helps.