It is common practice to use one's mouth as an additional "hand" to hold small nontoxic objects
Literally
mouthful of nails
might mean
but since your text has given he context of "He stayed at home and nailed up the mail slot."
It must be the first image.
"Cynical" is a confusing word because it has two meanings that are near opposites.
Merriam-Webster has these two descriptions:
Essential Meaning of cynical
1 : believing that people are generally
selfish and dishonest
She's become more cynical in her old age.
2 : selfish and dishonest in a way that shows no concern about treating other people fairly
Some people regard the governor's visit to the hospital as a cynical attempt to win votes.
The first definition is about the feeling or attitude someone has about someone else's behaviour. It's like skeptical with negativity.
The second definition is about the intent of someone's behaviour.
In all three of your examples above, "cynical" has the second definition. The writer is saying that US Government claimed they had the best interests of the Native Americans in mind, but really this was an excuse to justify serving their own interests at the expense of the Native Americans.
By contrast, today, the descendants of those Native Americans are cynical (in the first sense) that the government ever has their best interests in mind.
Another English word with a similar pair of meanings is "suspicious". Something can be suspicious, or a person can be suspicious of something. The two meanings of "cynical" parallel the two meanings of "suspicious".
Best Answer
A cloak is a full garment worn (usually outdoors) over all other clothes for warmth and protection. It is usually cut as a circle or partial circle, and is usually sleeveless but open at the front to give the hands and arms freedom. Cloaks are rarely worn in western cultures today; they have mostly been superseded by overcoats.
Robes are much more various. In mediaeval usage the word signified a sleeved garment with an ankle- or floor-length skirt, worn for non-strenuous activity; descendants of that sort of robe are usually rich, ornamented mantles worn over court dress on ceremonial occasions.
In more ordinary contexts a robe is a garment worn in very casual situations over swimsuits or pyjamas or nothing; it may be of very heavy or very flimsy fabric, depending on what activities are contemplated.
In Rowling's universe the word appeals to the mediaeval tradition, as a garment appropriate to sorcerers, but sober rather than flamboyant, like the gowns worn by present-day barristers, judges and scholars.