Context: The runaway star HD 34078, initially selected to investigate small scale structure in a foreground diffuse cloud, has been shown to be surrounded by highly excited H2, the origin of which is unclear. Aims: We first search for an association between the foreground cloud and HD 34078. Second, we extend previous investigations of temporal absorption line variations (CH, CH^+, H2) in order to better characterize them and understand their relation to small-scale structure in the molecular gas. Methods: We have mapped the 12CO(2-1) emission at 12'' resolution around HD 34078's position, using the 30 m IRAM antenna. The follow-up of CH and CH+ absorption lines has been extended over 5 more years: 26 visible spectra have been acquired since 2003 at high or intermediate resolution. In parallel, CH absorption towards the reddened star ζ Per has been monitored to check the instrumental stability and homogeneity of our measurements. Three more FUSE spectra have been obtained to search for N(H2) variations. Results: CO observations show a pronounced maximum near HD 34078's position, clearly indicating that the star and diffuse cloud are associated. The optical spectra confirm the reality of strong, rapid and correlated CH and CH+ fluctuations (up to 26% for N(CH^+) between 2007 and 2008). On the other hand, N(H2, J=0) has varied by less than 5% over 4 years, indicating the absence of marked density structure at scales below 100 AU. We also discard N(CH) variations towards ζ Per at scales less than 20 AU. Conclusions: Observational constraints from this work and from 24 μm dust emission appear to be consistent with H2 excitation but inconsistent with steady-state bow shock models and rather suggest that the shell of compressed gas surrounding HD 34078 or lying at the boundary of a small foreground clump is seen at an early stage of the interaction. The CH and CH+ time variations as well as their high abundances are likely due to chemical structure in the shocked gas layer located at the stellar wind/ambient cloud interface. Finally, the lack of variation in both N(H2, J=0) towards HD 34078 and N(CH) towards ζ Per suggests that quiescent molecular gas is not subject to pronounced small-scale structure. Based on observations made mainly at IRAM, Observatoire de Haute Provence (France), McDonald Observatory (USA) and with FUSE.