The definition is for "reprove" as a verb but in the example sentence, "self-reproving" is a noun. Usually, a person reproves another for a misdeed or a fault. But at this moment, Albany is scolding himself because he is not sure what to do: sit quietly while a foreign force invades his kingdom? or side with Goneril (whom he now hates) and Edmund (a bastard) against Cordelia and France who are fighting for Lear (whom he loves and respects)?
The definition is for the modern meaning of "forfend" as a verb. But in the example sentence, "forfended" is an adjective with the archaic meaning of "forbidden". Regan is questioning Edmund about Goneril. The audience knows from an earlier scene that Edmund is lying when he denies being intimate with Goneril, but the word "forfended" is another clue, since forbidden fruit is often the tastiest.
But have you never found my brother's way
To the forfended place?
Lear is trying to encourage Cordelia to be strong here. But his use of the word "devour" doesn't help, because the reason that he gives for Cordelia not to cry is that time will devour the flesh of her eyes so she should not feed her eyes with tears. Because Lear and Cordelia have been captured, they don't have the power to starve Goneril and Regan (who can also be "them"), but they can fight against their own eyes.
Wipe thine eyes.
The good-years shall
devour them, flesh and fell,
Ere they shall make us weep.
Another definition of "exalt" is "raise in rank, character, or status"--which is what Regan (legitimate daughter and heir of a king, and widow of a duke) did to Edmund (bastard son of an earl) by giving him the powers of her position. To undermine Regan, Goneril argues that Edmund does not need Regan to exalt him because he is already gloriously worth praising.
Not so hot.
In his own grace he doth
More than in your addition.
This definition doesn't actually fit the example sentence. Albany is using the adjective "capital" to describe treason as a crime that is "calling for the death penalty"--but for a crime to deserve that punishment, it must be seen as serious or of "primary importance".
an ulceration (especially of the lips or lining of the mouth)
Another definition of "canker" is "a pernicious and malign influence that is hard to get rid of"--which can describe the effects of treason on Edgar. But the given definition is more fitting for the example sentence because Edgar is using images of teeth and biting to describe the reason for his lost name.
Know my name is lost
By treason's tooth bare-gnawn and
come out better in a competition, race, or conflict
The definition is for "vanquish" as a verb, but "vanquished" is used as an adjective in the example sentence--this makes the meaning the opposite of what is given, since Goneril is arguing that Edmund was not conquered but cheated. This is a desperate attempt to restore honor to a fatally wounded Edmund who is lying at the feet of his opponent.
By the law of war, thou wast not bound to answer
An unknown opposite. Thou art not
but cozened and beguiled.
This is Albany's response to the news that Edmund is dead. Compared to the reported deaths of Gloucester, Regan and Goneril, the visible dead body of Cordelia, and the impending deaths of Kent and Lear, Edmund's death is but a trifle.