What is the ruling on offering condolences via newspapers and magazines and responding to them with thanks, and does this fall under the eulogizing (nayi) forbidden by Islamic law?

Yes, this may be considered part of the forbidden lamentation, and it involves extravagance and waste of money. Condolence is not like congratulation; its purpose is to strengthen the afflicted person to bear their calamity, not to be a courtesy or a congratulation. If people understood the purpose of condolence, they would not exaggerate in publishing it in newspapers or gathering for it and serving food.