Learn Italian: Sono talmente stanco da chiudere le palpebre per la sonnolenza. - I am so tired that I close my eyelids from drowsiness.

This is an implicit consecutive clause. When the subject of the main action (Sono stanco, I am tired) is the exact same as the subject of the resulting action (I close my eyelids), Italian pairs talmente with da followed by the infinitive (chiudere), rather than using che and a conjugated verb.

Could I use così instead of talmente in this sentence?

Absolutely. Sono così stanco da... and Sono talmente stanco da... are completely interchangeable and both mean exactly the same thing. Talmente can sometimes feel just a tiny bit more emphatic, but the grammar rules are identical.

Why does the sentence use per in per la sonnolenza to mean "from" drowsiness?

In Italian, the preposition per is the standard way to introduce a cause or reason. While English says "from" or "out of" in phrases like "from drowsiness" or "out of fear", Italian uses per, translating literally to "because of the drowsiness".

Is chiudere le palpebre the normal way to say you are closing your eyes to sleep?