Friction (and lubrication) between soft contacts is prevalent in many natural and engineered systems and plays a crucial role in determining their functionality. The contribution of viscoelastic hysteresis losses to friction in these systems has been well-established and defined for dry contacts; however, the influence of fluid viscosity and wetting on these components of friction has largely been overlooked. We provide systematic experimental evidence of the influence of lubricant viscosity and wetting on lubrication across multiple regimes within a viscoelastic contact. These effects are investigated for comparatively smooth and rough elastomeric contacts (PTFE-PDMS and PDMS-PDMS) lubricated by a series of Newtonian fluids with systematically controlled viscosity and static wetting properties, using a ball-on-disc tribometer. The distinct tribological behaviour, characterised generally by a decrease in the friction coefficient with increasing fluid viscosity and wettability, is explained in terms of lubricant dewetting and squeeze-out dynamics and their impact on multi-scale viscoelastic dissipation mechanisms at the bulk-, asperity-, sub-asperity- and molecular-scale. It is proposed that lubrication within the (non-molecularly) smooth contact is governed by localised fluid entrapment and molecular-scale (interfacial) viscoelastic effects, while additional rubber hysteresis stimulated by fluid-asperity interactions, combined with rapid fluid drainage at low speeds within the rough contact, alter the general shape of the Stribeck curve. This fluid viscosity effect is in some agreement with theoretical predictions. Conventional methods for analysing and interpreting tribological data, which typically involve scaling sliding velocity with lubricant viscosity, need to be revised for viscoelastic contacts with consideration of these indirect viscosity effects.