Learn Finnish: Kenelle sinä lähetät tämän paketin?

Kenelle is the form of kuka (who) with the -lle ending added to it. The -lle ending means "to" or "onto" someone. So, just as minulle means "to me", kenelle means "to whom".

Why do the words tämä (this) and paketti (package) end in -n here?

They make up the object of the sentence (the thing being sent). In Finnish, when you do an action to a whole object, you usually add an -n to the end of it. Notice that both the noun (paketin) and its describing word (tämän) must match and take the same ending.

Why does lähetät have only one t in the middle, when the dictionary form is lähettää?

This happens because of a rule called consonant gradation. When you add the "you" ending (-t) to the verb stem, it closes off the final syllable (-tät). This "closed door" forces the strong double tt to weaken into a single t.

I noticed paketin also dropped a t compared to its basic form paketti. Is that the exact same rule?

Yes, exactly! Just like with the verb, adding the object ending (-n) to paketti closes the final syllable (-tin). This triggers the same consonant gradation rule, weakening the strong tt into a single t.