It was recently shown that the exponentially repulsive EXP pair potential defines a system of particles in terms of which simple liquids' quasiuniversality may be explained [A. K. Bacher et al., Nat. Commun. 5, 5424 (2014); J. C. Dyre, J. Phys.: Condens. Matter 28, 323001 (2016)]. This paper and its companion [A. K. Bacher et al., J. Chem. Phys. 149, 114502 (2018)] present a detailed simulation study of the EXP system. Here we study how structure monitored by the radial distribution function and dynamics monitored by the mean-square displacement as a function of time evolve along the system's isotherms and isochores. The focus is on the gas and liquid phases, which are distinguished pragmatically by the absence or presence of a minimum in the radial distribution function above its first maximum. A constant-potential-energy (NVU)-based proof of quasiuniversality is presented, and quasiuniversality is illustrated by showing that the structure of the Lennard-Jones system at four state points is well approximated by those of EXP pair-potential systems with the same reduced diffusion constant. Paper II studies the EXP system's isomorphs, focusing also on the gas and liquid phases.