Yes, the parenthesis at least is correct.
Yes, the usage of "speaks" is correct in that sentence. This is more obvious when one removes all the grammatically irrelevant other stuff between the verb in question and the noun that goes with it.
The subject is "a subordinate" and it has two verb clauses, "makes..." and "speaks..." separated by the conjunction "and". If we strike the first clause, we get:
a subordinate usually speaks with sincere or, at least, neutral tone in order to avoid offending.
Dropping the adverb "usually" and disregarding everything else that modifies "speak", we get:
a subordinate speaks
which is exactly correct. Singular noun, present tense verb is stem+"s".
I think maybe your confusion was that you were anticipating the second clause to be in reference to the verb expressed. Maybe you were headed for something like:
Suggesting expressions are usually expressed when a subordinate makes a proposal or suggestion to the superior and are usually spoken with sincere or, at least, neutral tone in order to avoid offending.
By sticking that "are" in there, we've changed the grammatical structure of the sentence. No longer does the "and" separate two clauses that describe what the subordinate does; it now separates two clauses that describe how suggesting expressions are usually expressed. The "and" is like the fork in a tree branch; you can tell where the new branch is along the old one by its grammatical structure, which is interpreted as strictly parallel with the original.
Since "speaks" is in the same form as "makes", it branches off right before "makes". By sticking an "are (usually) spoken" in there, we make it parallel with "are (usually) expressed", so that's now where the sentence forks.