Hardware synthesis from high-level descriptions remains fundamentally limited by the sequential optimization of interdependent design decisions. Current methodologies, including state-of-the-art high-level synthesis (HLS) tools, artificially separate implementation selection from scheduling, leading to suboptimal designs that cannot fully exploit modern FPGA heterogeneous architectures. Implementation selection is typically performed by ad-hoc pattern matching on operations, a process that does not consider the impact on scheduling. Subsequently, scheduling algorithms operate on fixed selection solutions with inaccurate delay estimates, which misses critical optimization opportunities from appropriately configured FPGA blocks like DSP slices. We present SkyEgg, a novel hardware synthesis framework that jointly optimizes implementation selection and scheduling using the e-graph data structure. Our key insight is that both algebraic transformations and hardware implementation choices can be uniformly represented as rewrite rules within an e-graph, modeling the complete design space of implementation candidates to be selected and scheduled together. First, SkyEgg constructs an e-graph from the input program. It then applies both algebraic and implementation rewrites through equality saturation. Finally, it formulates the joint optimization as a mixed-integer linear programming (MILP) problem on the saturated e-graph. We provide both exact MILP solving and an efficient ASAP heuristic for scalable synthesis. Our evaluation on benchmarks from diverse applications targeting Xilinx Kintex UltraScale+ FPGAs demonstrates that SkyEgg achieves an average speedup of 3.01x over Vitis HLS, with improvements up to 5.22x for complex expressions.