In this paper, we show how a decomposition of a free choice Petri net into a ``routing'' network and marked graph subnetworks (i.e. linear subnetworks in the $[\max, +]$ setting) leads to new methods and algorithms to test structural as well as temporal properties of the net. Although several results hold for general free choice nets, the paper primalily focuses on the class of single input-free choice nets, defined here. We show how this decomposition in linear subnets allows one to: (in the untimed case) check liveness in polynomial time; (in the timed case) establish evolution equations which allow to represent the system as a coupling of two linear systems, a $(\min,+)$-linear system, and a quasi $(+,\times)$-linear one; (in the stochastic case) check stability, i.e. the fact that the marking remains bounded in probability. The main tools for proving these properties are graph theory, idempotent algebras and ergodic theory.