Learn Polish: Czy masz jeszcze swój stary plecak szkolny? - Do you still have your old school backpack?

Szkolny is a relational adjective meaning school-related. To form it, we drop the final -a from szkoła and add the suffix -ny. Notice the spelling change: the hard ł softens into an l before the suffix, resulting in szkolny.

Why is stary placed before the noun, but szkolny is placed after it?

This is a great example of Polish adjective placement rules. Stary (old) is a descriptive adjective that tells you a quality, so it goes before the noun. Szkolny (school) is a relational adjective that categorizes the noun into a specific type (a school backpack, not a hiking backpack). Classifying adjectives almost always go after the noun.

Could I say twój instead of swój here?

While you might hear Polish speakers use twój in casual conversation here, swój is the grammatically correct choice. Because the owner of the backpack is the same as the subject of the sentence (you), you should use the reflexive possessive pronoun swój.

Does jeszcze mean still, yet, or more in this sentence?

Here, it means still, inquiring if an ongoing situation is continuing. You will usually only see jeszcze translated as yet in negative sentences (like jeszcze nie), and as more when dealing with amounts or additions (like jeszcze jeden, meaning one more).