The increasing popularity of compute acceleration for emerging domains such as artificial intelligence and computer vision has led to the growing need for domain-specific accelerators, often implemented as specialized processors that execute a set of domain-optimized instructions. The ability to rapidly explore (1) various possibilities of the customized instruction set, and (2) its corresponding micro-architectural features is critical to achieve the best quality-of-results (QoRs). However, this ability is frequently hindered by the manual design process at the register transfer level (RTL). Such an RTL-based methodology is often expensive and slow to react when the design specifications change at the instruction-set level and/or micro-architectural level. We address this deficiency in domain-specific processor design with ASSIST, a behavior-level synthesis framework for RISC-V processors. From an untimed functional instruction set description, ASSIST generates a spectrum of RISC-V processors implementing varying micro-architectural design choices, which enables effective tradeoffs between different QoR metrics. We demonstrate the automatic synthesis of more than 60 in-order processor implementations with varying pipeline structures from the RISC-V 32I instruction set, some of which dominate the manually optimized counterparts in the area-performance Pareto frontier. In addition, we propose an autotuning-based approach for optimizing the implementations under a given performance constraint and the technology target. We further present case studies of synthesizing various custom instruction extensions and customized instruction sets for cryptography and machine learning applications.