What is the ruling on carrying the Mus'haf while there is water on one's hands, or placing it on a book or table that has dust on it? And is it obligatory to wipe the dust off it before carrying it to avoid its desecration?

There is no harm in carrying the Mus-haf with wet hands, as long as it does not lead to damage to its pages or obscuring its letters. Placing the Mus-haf on a dusty table is not considered a desecration, and one is not required to remove all dust due to the hardship involved. However, doing so out of reverence for the Mus-haf is good, for Allah Almighty says: "That [is so]. And whoever honors the symbols of Allah — indeed, it is from the piety of hearts." Wiping dust off it is among the commendable ways of honoring and respecting it. One should avoid falling into obsessive doubts regarding these matters.