What should I write for "Research and Teaching Merits" when applying online for a PhD?
AFAIK, the section Research and Teaching Merits ought to encompass accomplishments made as a researcher and as a teacher. This could include any distinguished teaching award you might have received in your teaching tenure. If there isn't a separate publication field, you may include your publication here.

As for the field named Field of research, this would mean your current domain within the main subject of research. For instance, in Computer Science, fields could be like Machine Learning, Computational Neurosciences, Wireless Sensor Networks, Data Security, etc. Likewise, there would be many specialization areas within Mathematics, Electronics, Instrumentation, Mechanics, and so on.