1. Lets assume that there is a requirement for the "ba'al" to be mentally healthy at the time the "shaliach" gives the "geth". How do we confirm it? It could take a month of traveling for the "shaliach" and a lot could change in this month.
In addition, what happens if the "baal" dies during that month. How do we confirm or synchronize his required state? By the way it looks very similar to a problem the communication engineers have when they want to be sure that a message arrived OK.
2. Regarding the mute not speaking. He actually can speak, but we cannot hear. In addition we could read his lips and this would be coming from his mouth.
3. Could a mute have a "shaliach" for testimony that would act as his mouth?
This would be similar to a person that has no hands and cannot "lehagbiah" a
"metziah" and asks a "shaliach" to make the "hagbaha" for him.
1. A get is given on the assumption that the sender is alive and sane at the time that it arrives. We rely on "Chazakah." Chazakah teaches us to assume that the physical status of the husband has not changed unless we know otherwise. (See Gitin 29a.)
2. As for the mute, "Cheresh" refers to the deaf-mute, who is assumed to have a weak intellect (lack of "Da'as," like a minor) due to his inability to communicate properly. He is not simply unable to make himself heard. An "Ilem," or mute who can hear, is considered to be a "Bar Da'as," who merely has a physical handicap. Reading his lips would be a form of Remizah; however it would not be considered speech ("mi'Pihem"). Speech must be heard.
3. Testimony must be delivered by the person's own speech, and not by a Shali'ach. Even written testimony is invalid. The obvious reason (although I am oversimplifying ) is that if we rely on a Shaliach, we are believing the Shaliach rather than the Ed.