Learn Finnish: Opitko uusia sanoja lukemalla tekstejä? - Do you learn new words by reading texts?

This uses the third infinitive adessive form, which expresses how or by what means something is done. You take the strong stem of the verb lukea (which is luke-), add the third infinitive marker -ma-, and finally add the adessive case ending -lla.

Why are uusia sanoja and tekstejä in the partitive plural case?

Both function as indefinite plural objects in this sentence. You are asking about learning "some new words" by reading "some texts" rather than specific, previously mentioned words and texts. In Finnish, an indefinite plural object takes the partitive case.

Since lukemalla is an infinitive here, how does it take tekstejä as an object?

In Finnish, infinitive verb forms can still take their own objects just like conjugated verbs do. The word tekstejä acts as the direct object of the action "reading" and follows normal object rules, which is why it is in the partitive plural.

Could the question particle -ko be attached to a different word here?

Yes, but it changes the focus of the question. Attached to opitko (Do you learn...?), it asks a general yes/no question. If you wanted to emphasize the method, you could move the particle and ask Lukemallako opit uusia sanoja? which means "Is it by reading that you learn new words?"