Is it permissible to expose the nakedness of a father suffering from Alzheimer's to clean him? And is it permissible to let him go to the mosque alone despite fearing for him, or to wait for him to finish his ablution for congregational prayer?

If the father's intellect is gone, such that he does not understand speech nor comprehend answers, then the burden of religious obligation (taklif) is lifted from him. However, if he is of sound mind, then he remains obligated, and you must treat him kindly and attend to his well-being. It is permissible to look at his private parts for cleaning if he is unable to do so himself, due to necessity. If you miss congregational prayer at the mosque because you are attending to him, there is no blame upon you, but you should strive to perform congregational prayer, even if at home.