Learn Finnish: Olen väsyneempi, koska en voinut torkkua lainkaan. - I am more tired, because I could not doze at all.

Both words mean "at all" and they are usually completely interchangeable. You could easily say en voinut torkkua ollenkaan here. Lainkaan is just slightly more formal and often used in written Finnish, but you will hear both in everyday speech.

Does lainkaan have to go at the very end of the clause?

Usually, yes. Negative polarity adverbs like lainkaan and ollenkaan strongly tend to appear at the end of the clause they modify. They wrap up the negative statement, so placing it anywhere else in en voinut torkkua lainkaan would sound unnatural.

Can lainkaan ever be used in a positive sentence?

No, it requires a negative word like en in the same clause, or at least a strongly negative context (like a rhetorical question). You cannot use it to say something positive like "I managed to doze at all".

How exactly does the adjective väsynyt become the comparative väsyneempi?