Is this ceiling fan going to fall and cut off my head?
Is this ceiling fan going to fall? There is nothing in the photo that tells me a fall is about to happen, but the installation looks poor, and I think you need to get under that cover.
Will it cut your head off? No, if the physical support let’s go, it will dangle by it’s wiring like a wounded buzzard, but probably give you time to get clear. A bonk with a thin piece of wood is the worst you are in for.
Would I trust this fan? No. The installation is poor and I’m concerned the true support (the socket) is also poorly installed. The cover you see does not provide support.

Ceiling fans use a ball and socket system, the ball is what you see in your photo. The socket is under the cover and it is usually screwed into a connection box, and that connection box should be firmly supported. The fan is heavy.
How can you tell if it’s secure? That cover should unscrew and drop down. That will let you see how the socket is supported.
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Ok, sounds like you are not getting a solid hold with the screws. A retrofit (or old work) ceiling fan brace is the way to go. This video shows you the process, don’t have to buy theirs, there are a lot to choose from. You will have to take the fan down and disconnect it from the wiring, but that’s probably a good idea at this point, kind of a mess up there.
https://youtu.be/Hf-qKY2bRlo