Edge computing brings computation closer to Internet-of-Things (IoT) data sources, reducing latency but increasing energy consumption and susceptibility to node failures. The cloud platform provides extensive computational capabilities, but comes with significant costs and communication delays due to network congestion. The edge-cloud continuum strategically combines these approaches to mitigate their individual drawbacks. However, effectively scheduling IoT workflows to minimize costs while adhering to strict requirements for latency, energy efficiency, and reliability remains a major challenge in real-time IoT applications. To address these challenges, we propose the Reliable Energy-constrained Cost-aware Real-time (RECR) algorithm for optimizing IoT workflow scheduling across the edge-cloud continuum. RECR minimizes monetary costs and enhances reliability while adhering to strict energy and deadline constraints. We also introduce RECR-D, a fault-tolerant extension that employs adaptive task duplication to manage transient and permanent failures, with reliability rigorously modeled using Continuous-Time Markov Chains (CTMCs) to integrate dynamic failure behavior. Extensive simulations demonstrate that RECR reduces workflow monetary costs by approximately 21% and improves deadline adherence by 37% compared to state-of-the-art algorithms. Furthermore, RECR-D improves compliance with reliability and energy constraints by 27% and by up to 208%, respectively, highlighting its robust performance in dynamic, failure-prone environments. These contributions significantly advance workflow management for IoT applications, proving crucial for real-time traffic control and video analytics in smart cities, ensuring timely processing and lower costs. They are also vital for remote patient monitoring and medical imaging analysis in healthcare, improving reliability and meeting deadlines for patient safety.