Does the reward for praying Fajr immediately after the adhan differ from the reward for praying it before sunrise, and is praying late, just before sunrise, considered delaying it or does it diminish its reward?

Initiating prayer at the beginning of its time is among the best deeds. The reward for prayer is greater if it is performed immediately after the adhan (after the Sunnah prayer). This does not mean it is obligatory to perform it upon hearing the adhan; rather, prayer is permissible throughout the entire time. Sin is incurred by whoever deliberately delays it beyond its chosen time.