• This translation may not be perfect, but is its general sense correct? (I.e. the speaker believes that none of the listeners understands.)

Je doute is a little bit ambiguous.
If you say "Je doute qu’aucun d’eux ne comprenne ton discours" it may be understood as the speaker does not believe in the fact that none of the listeners understands, i.e. he believes that at least someone understood. Je doute is like Je ne pense pas. To avoid ambiguity, you can say

Je pense qu’aucun d’entre eux ne comprend ton discours.

• how would one say this in French?: I doubt any of them does not understand what you’re saying.

Linked with the previous sentence, you could translate it this way:

Je ne pense pas qu’aucun d’entre eux ne comprend ton discours.

Or, to avoid double negation:

Je pense qu’ils comprennent tous ton discours.