Czyje są te ołówki przy oknie?

Because the interrogative possessive must agree with the grammatical gender and number of the possessed noun, not with the owner. Here the possessed noun is ołówki (pencils), which are plural, non‑masculine‑personal. In the plural:

• masculine personal (groups of men): czyi
• all other plurals (masculine inanimate, neuter, feminine): czyje

So we say czyje (ołówki). For comparison:

• singular masculine: czyj (ołówek)
• singular feminine: czyja (książka)
• singular neuter: czyje (okno)

In the plural, the demonstrative pronoun distinguishes only masculine personal vs. everything else:

• masculine personal plural (groups of men): ci (e.g., ci chłopcy)
• all other plural nouns: te (e.g., te ołówki, te książki, te okna)

Since pencils are not people, it’s te ołówki.

Yes. Both are correct:

• Czyje są te ołówki przy oknie? (fully explicit copula + demonstrative)
• Czyje to ołówki (przy oknie)? (very natural, common in speech) The second is a bit snappier and very idiomatic.

The preposition przy (by/next to) governs the locative case. Neuter okno in locative singular is oknie. Other examples:

• przy biurku (by the desk)
• przy drzwiach (by the door, plural noun)

Yes:

• obok okna, koło okna (next to/near the window) take the genitive (okna), not locative.
• w oknie (in/at the window opening) takes the locative (oknie) but means something different.
• na oknie literally “on the window” and is used for the window surface/sill contextually; more precise is na parapecie (on the windowsill).

Many masculine nouns ending in -ek drop the -e- in the plural. With a stem ending in k/g, the plural typically gets -i:

• ołówek → ołówki
• kwiatek → kwiatki
• domek → domki Key case forms for reference:
• nominative plural: ołówki (subject)
• genitive plural: ołówków
• locative plural: ołówkach
• instrumental plural: ołówkami

No, it’s optional. Compare:

• Czyje są te ołówki przy oknie? = “Whose are these pencils by the window?” (points to a specific set)
• Czyje są ołówki przy oknie? = “Whose are the pencils by the window?” (could be more general/identifying which owner)

They look similar but function differently:

• czy introduces yes/no questions: Czy te ołówki są twoje?
• czyje is the possessive interrogative “whose,” agreeing with the noun: Czyje są te ołówki? There’s no czy particle in your sentence.

Prefer czyj/czyja/czyje/czyi before a noun. Kogo is the genitive of “kto” (who) and is used in structures like:

• Do kogo należą te ołówki? (To whom do these pencils belong?) Using kogo directly before a noun (e.g., “kogo ołówki?”) is nonstandard or very colloquial; stick with czyje ołówki? in standard Polish.

Polish word order is flexible, but some versions sound more natural:

• Very natural: Czyje to ołówki przy oknie?
• Also fine: Czyje są te ołówki przy oknie?
• Less natural/marked: Te ołówki przy oknie są czyje? (rhetorical/emphatic) Keep the wh-word (czyje) early for a straightforward question.

The interrogative czyj declines. Examples:

• Accusative plural (no change in form): Czyje ołówki widzisz? (Which/whose pencils do you see?)
• Locative plural: O czyich ołówkach mówisz? (About whose pencils are you speaking?)
• Genitive singular masculine: Z czyjego ołówka korzystasz? (Whose pencil are you using?) So the ending of “czyj–” adapts to the case required by the sentence.

Polish doesn’t mirror that English structure naturally. The preferred pattern is to front the interrogative:

• Czyje są te ołówki? or, more idiomatically, Czyje to ołówki? Ending a sentence with czyje sounds marked or unnatural outside specific rhetorical contexts.