Learn Hardcore Turkish: Çayı bardağa dök, lütfen. - Please pour the tea into the glass.

dök: pour (tip/pour out); can sound a bit brisk as a command.
koy: put; in everyday Turkish, also used for pouring into a container (e.g., barda <a class="border-b border-dotted border-blue-400 dark:border-blue-500 cursor-pointer text-inherit no-underline" href="/grammar/turkish/alphabet/g-yumusak">ğ a su/çay koy is very common).
doldur: fill (emphasizes filling up a container).
boşalt: empty/pour out completely (focus on emptying the source). All can be correct depending on nuance; for serving tea, koy and doldur are very common.

No. The root is çay, which already ends in y. You’re just adding the accusative -ı: çay-ı → çayı. A buffer y appears only when the noun ends in a vowel (e.g., su → suyu “the water”).

What does bardağa mean morphologically, and why does k turn into ğ?

Base noun: bardak “glass”.
Dative (to/into): -(y)a / -(y)e → last vowel is back (a), so choose -a.
Many nouns ending in -k soften to -ğ- before a vowel-initial suffix: bardak + a → bardağa. Note: Not every -k becomes -ğ; some become -g (e.g., renk → rengi). This is a common but not universal alternation.

Why dative bardağa and not locative bardakta?