Learn Hardcore French: Paul chante le mieux quand il est assis.

Because mieux/le mieux is the comparative/superlative form of the adverb bien, and it modifies a verb (here, chante). Meilleur/le meilleur is the comparative/superlative form of the adjective bon, and it modifies a noun. So you say Il chante le mieux, but C’est le meilleur chanteur.

Should it be la mieux if the subject is feminine?

No. With adverbs, the superlative uses the invariable article le. You say Elle chante le mieux, not la mieux.

Where does le mieux go in the sentence?

With simple tenses, adverbs of manner like this typically go after the verb: Paul chante le mieux. In compound tenses, they usually go after the past participle: Il a chanté le mieux.

Can I move the quand-clause to the beginning, and do I need a comma?

Yes: Quand il est assis, Paul chante le mieux. When the quand-clause comes first, put a comma after it. When it follows the main clause, you normally don’t use a comma: Paul chante le mieux quand il est assis.

Why is it assis and not assise? How does agreement work?

Assis is the past participle of asseoir used as an adjective meaning seated/sitting. It agrees with the subject:

masculine singular: assis (Paul est assis)
feminine singular: assise (Marie est assise)
masculine plural: assis
feminine plural: assises

Is there a difference between être assis and s’asseoir?

Yes. Être assis describes a state (being seated). S’asseoir describes the action of sitting down. So quand il est assis = when he is seated (state), whereas quand il s’assoit/s’assied = when he sits down (the act). Both il s’assoit and il s’assied are standard variants.

Does le mieux here mean “best compared to others” or “best for Paul compared to other times”?

In this sentence, the quand-clause narrows the comparison to Paul’s different conditions: he sings best when seated (as opposed to when standing, moving, etc.). To make a group comparison explicit, you could add something like de tous: Paul chante le mieux de tous quand il est assis.

Can I say au mieux instead of le mieux?

No. Au mieux means at best/to the best of his ability (a limit), not the superlative “the best.” Il chante au mieux suggests “he does the best he can,” not “he sings the best (of all conditions/people).”

Can I shorten it to Paul chante le mieux assis?

Yes. This is a common, concise alternative where assis functions as a predicative adjective referring back to the subject: Paul chante le mieux assis. For a woman: Marie chante le mieux assise.

Can I use lorsque instead of quand?

Yes. Lorsque is a bit more formal or literary but fully correct: Paul chante le mieux lorsqu’il est assis. Note the elision: lorsqu’il before a vowel sound.

Which tenses should I use with quand for future situations?

French uses the future in both clauses: Quand il sera assis, il chantera le mieux. For habitual facts, stick with the present in both: Quand il est assis, il chante le mieux.

How do you pronounce the tricky parts, and are there liaisons?

mieux is pronounced roughly like [mjø] (one syllable).
assis is [a-si]; the final -s is silent unless there’s liaison after it.
There’s a required liaison between est and assis: il est assis → [il ɛt‿asi].
A liaison is common after quand before a vowel: quand il → [kɑ̃.t‿il].