What do you mean by "same spot"? Relative to what? The sun? The galactic center?

I'm going to assume you mean the standard "what would happen if I hovered in a helicopter for hours, would I be somewhere else?"

No. You are moving right now. When a helicopter (or you) hovers, it is matching the rotational speed of the ground below it. Let's say what you mean is that you take off at sunrise and for 12 hours you want to keep the sun at the same visual spot on the horizon. You would no longer be hovering because you would have to counter your current speed, meaning you would have to race West at something like 1000 miles an hour. You have to counter the rotational speed of the earth at your latitude and I believe if you do this at sunrise, you don't have to add the Earth's speed around the sun, you just have to maintain your elevation because the earth is moving toward you as it sounds beneath you.

But I think OP's question is why did he just accept that she was there? As OP said, her surviving his attempt was conceivable, but what did he think was going to happen when she apparently survived and showed up on the island? She'd just be cool? Not say anything? Live and let live?

I'm guessing he's just flat out too stupid to come up with a response. Maybe if "Andi" had confronted him he'd come up with something in the moment to play it off, but couldn't think of his own move since he's pretty much only reactive. He knows Duke saw him but doesn't do anything or have any plan until he's blackmailed and makes a panicked move.

Maybe he could've made a scene and tried to send her away when she appears at the beach; he'd have to explain why she got an invite, but sometimes people invite others with the expectation it'll be turned down, so I don't think it'd be so crazy. The shitheads probably wouldn't challenge him since they themselves said she shouldn't have come. He'd sent everyone maintaining the property away so it's not like he could've had someone watch her.

It's shown. You can hear the glass retract. That's how it keeps closing over and over every time Duke's phone dings.

Enemy of the State

Probably won't see it, especially if it's not on a service I already have. The commercials didn't do anything for me; I wasn't sure it was real because Ryan and Will in a Carol remake seemed like a strange concept.

Only Carols I watch for Christmas are Scrooged and The Muppets.

Kept looking through TV Tropes, didn't see anything on The Specialist, so started looking for myself.

Plot-Sensitive Snooping Skills

Maybe Failed a Spot Check, Selective Obliviousness

I would guess it's not the dystopian fascist state he has a problem with, having grown up in it and rose in the ranks of the group that keeps that state in power, but the shooting of unarmed citizens, which this must've been the first time he himself was ordered to do so. We know they suppress all news so it wouldn't be out of the question that Ben didn't know the murder of unarmed civilians was something that happened.

I took a look at The Running Man's TV Tropes but nothing jumped out at me as what you're looking for.

Came to mention 65. It's maddening. There's a sign on the wall near my outside mailbox kiosk thing. "This area may contain..." Can you be more specific? Is there something I should be looking out for? Are you just telling me not to eat the plants or chew on the stucco? Whatever good intention there may have been has been obliterated by people putting these warnings on everything to cover their asses.