Learn Turkish: Raporu okuyoruz. - We are reading the report.

The -u is the accusative case suffix. It marks the word as a specific direct object. Without it, rapor would just mean 'a report' or general report-reading. With it, raporu means 'the report'.

If I want to ask 'What are we reading?' to get this answer, do I use ne or neyi?

You would use neyi (what specific thing). Because raporu has the accusative suffix showing it is a specific report, your question must also match that specificity by asking Neyi okuyoruz?

Why does rapor take -u for its specific object ending instead of -i or -ı?

Turkish uses 4-way vowel harmony for this suffix. Since the last vowel in rapor is 'o', the harmony rules dictate that the suffix must be -u.

The verb root for 'to read' is oku. How is the present tense built here?

The present continuous suffix is -yor. Usually, a connecting vowel is needed if a verb ends in a consonant. But since oku already ends in a vowel, we just add -yor directly, followed by the 'we' ending -uz, giving us okuyoruz.