Mental health challenges among Indian adolescents are shaped by unique cultural and systemic barriers, including high social stigma and limited professional support. Through a mixed-methods study involving a survey of 278 adolescents and follow-up interviews with 12 participants, we explore how adolescents perceive mental health challenges and interact with digital tools. Quantitative results highlight low self-stigma but significant social stigma, a preference for text over voice interactions, and low utilization of mental health apps but high smartphone access. Our qualitative findings reveal that while adolescents value privacy, emotional support, and localized content in mental health tools, existing chatbots lack personalization and cultural relevance. These findings inform recommendations for culturally sensitive chatbot design that prioritizes anonymity, tailored support, and localized resources to better meet the needs of adolescents in India. This work advances culturally sensitive chatbot design by centering underrepresented populations, addressing critical gaps in accessibility and support for adolescents in India.