Is it possible for an Alzheimer’s/demented person to fall purposely to “get attention"?

People with Alzheimer's are inherently unstable. Even using a walker may not help much. Falling is a serious issue. If a bone is broken, it could accelerate the decline in someone. People with Alzheimer's do not heal well. Those with AD can be very resourceful, but I don't think they seek attention, instability is part of Alzheimer's.

Yes it is. Trust your instincts if you think it's happening. Pick them up, but don't get too over-the-top with your attention. My dreadful father would fake anything. Brain damage or not, people can be very crafty.

People who have dementia have brain damage. They may function better at some times than other times. I’m not sure how you could determine that a person with dementia is acting recklessly on purpose. If they are hurting themselves, they need protection and safeguarding, regardless of the reasons. I’d discuss it with his doctor. People with dementia require care and attention at all times, even if they are not falling.