My guess would be predatory birds, but I don't know much about the animals you have there. My place has a lot of natural areas with bush, swamp and creek so there's a great variety of wildlife here. One of the main reasons I got this place. Whilst I'm a keen observer (and photographer) of the wildlife I know that for every animal I see a lot of other animals died to keep it alive to that point. The reason there are so many animals here is that there's so many animals for them to eat. It's the reality of nature.

On the other side of your barricade, is there a natural fresh or saltwater water front in the near vicinity? If so, you can add mink to the long list of possibilities. I had to put up a game camera in my yard to catch sight of the culprits; I had no idea that they existed on my property.

Setting up a camera is what came to my mind also.

"is there a natural fresh or saltwater water front in the near vicinity?"

Yes, I am in Fort Lauderdale Florida, and I am within 20 feet from the Tarpon River, there is a Marina across the street where people launch their boats and park their yachts. I don't know if the water in the river is fresh or salty as we are 1 mile from the ocean so I am thinking it's brackish water?

I am not sure there are minks down here. Really? May be weasels?

"My guess would be predatory birds"

Could birds tear apart a large crab and eat it? We have lots of ducks around, since we are so close to the river, but they cannot get over the fence. I don't think they would attack lizards either.

There are two other kinds of birds around not sure if they may be capable. They are around the outside but I have not seen them inside my fence though.

The long beak ones look predatory.

The one in your first picture is a duck or goose. The others are spoonbills. Surely there are a lot more kinds of birds around there than ducks/geese and spoonbills - tropic was probably thinking of hawks for instance.

Anyway somebody at the nearest Department of Wildlife office probably knows what would leave behind random parts in your area.

Our ducks normally fly around which I thought was normal for all ducks. They walk looking for food but readily fly when they want to. Your long beaked birds look like Ibis of some kind which can demolish a crab.

Crabs have an exoskeleton (outside of body) which splits when the crab outgrows it. They wriggle out of the exoskeleton and their new one underneath is very soft making them vulnerable to a lot of predators. It takes a while to harden.

In relation to Anoles, I was thinking of predatory birds such as raptors (hawks and the like). Presumeably Anoles roost in trees at night so they would also be more vulnerable to owls. All these birds tear their prey apart to eat it and a head could easily be dropped or left behind.

Possibly owls. I don't think a hawk is a possibility because my property has so much tree canopy there isn't much open space.

Can I rule out snakes? A snake would have swallowed the prey whole right?

Yes, I'd rule out snakes for the reason you gave.

We have some hawks that are ambush predators. They 'sneak" around in the foliage of trees and pounce on their prey, both in the trees and on the ground. You may have some like that. They're not seen all that much. They have "eyes like hawks" and see you before you see them and move off.

It looked before like the two wading birds had broad ends to their bills.

Embothrium, I thought that as well at first but then decided the bills might be open giving that impression. I don't know of any Spoonbills with those dark colours.

They have sharp bills.

Definitely Ibis but which specific one is more difficult to tell. Possibly juvenile White-faced Ibis, but difficult to tell apart from a juvenile Glossy Ibis. Crabs are definitely on their menu. Lizards and snakes they'd swallow whole.

Miami, yes....minks are a possibility in your area. I'll bet the the culprit is a neighborhood or feral cat. They will commonly decapitate their victims, just for the heck of it.

It would be because they don't want the head.

There are a bunch of feral cats on the same street. I am fully fenced in though but I have seen one of the cats twice inside my property watching the goldfish in my pond. As soon as the cat saw me he jumped to the top of the six foot fence and ran away.

Tonight at dusk I saw another critter. I wouldn't even see it but I happened to be coming home after dark and had my headlights on and I saw what looks to be a big rat the size I would say 12" to 14" in length walking across the driveway. It was dark but I think it's a possum.