Quantum simulation is the foundation for the design of many algorithms which share subroutines known as quantum simulation kernels. Optimizing the compilation of these kernels is crucial, involving two key components: 1) circuit synthesis and 2) qubit mapping. However, existing circuit synthesis methods either overlook qubit connectivity constraints (QCCs) or prioritize minimizing gate count over optimizing circuit depth. Similarly, current qubit mapping techniques do not work well with circuit synthesis methods. To address these limitations, we propose PauliForest, which comprises a connectivity-aware circuit synthesis algorithm and a Pauli-oriented qubit mapping algorithm. The synthesis algorithm employs heuristic strategies to generate shallower circuits, while the qubit mapping algorithm seamlessly collaborates with the circuit synthesis process. Compared to the state-of-the-art Paulihedral compiler, our approach significantly reduces both CNOT gate counts (by 13%) and circuit depths (by 25%). Experiments on a noisy simulator and a real superconducting quantum computer show that our algorithm can improve the fidelity of quantum circuit execution compared to Paulihedral.