Learn Turkish: Ben genel kategoriyi seçtim. - I chose the general category.

In Turkish, the most important new information (the focus) goes immediately before the verb. Here, genel kategoriyi is in that pre-verbal focus position, emphasizing exactly what it was that you chose.

If I wanted to emphasize that I chose it (and not someone else), how would the sentence change?

You would move the subject ben into the pre-verbal focus spot! The sentence would become Genel kategoriyi ben seçtim. In our current sentence, ben sits at the beginning simply as the topic, while the category is the focus.

Why does kategori have the -(y)i ending here?

It marks the specific direct object (the accusative case). Because you are choosing a specific category, we add the buffer -y- and the vowel -i to show that it is the specific target receiving the action of the verb seçmek.

Why does the past tense suffix on seçtim start with a t instead of a d?

This is due to consonant harmony. The verb root seç- ends in a voiceless consonant (ç). When you add the past tense suffix -di, the d must change to t to match the voiceless sound, making it -ti before adding the -m for the first person.