The quest for low-cost embedded AI/ML applications has motivated innovations across multiple abstractions of the computation stack. Novel approaches for arithmetic operations have primarily involved quantization, precision-scaling, approximations, and modified data representation. In this context, Posit has emerged as an alternative to the IEEE-754 standard as it offers multiple benefits, primarily due to its dynamic range and tapered precision. However, the implementation of Posit arithmetic operations tends to result in high resource utilization and power dissipation. Consequently, recent works have delved into the idea of exploiting the error resilience of machine learning algorithms by using low-precision Posit arithmetic. However, limiting the exploration to precision-scaling limits the scope for application-specific optimizations for embedded AI/ML applications. To this end, we explore operator-level optimizations and approximations for low-precision Posit numbers. Specifically, we identify and eliminate redundant operations in state-of-the-art Posit arithmetic operator designs and provide a modular framework for exploring approximations in various stages of the computation. We also present a novel framework for behaviorally testing the corresponding Posit approximate designs in Artificial Neural Networks. The proposed optimizations and approximations exhibit considerable resource improvements with a small error in many cases. For instance, a Posit-based multiplier with 1-bit reduced precision shows a 33% improvement in power and utilization, with only a 0.2% degradation in overall accuracy.