Does a turboshaft's exhaust provide any thrust?
Yes, the exhaust of any engine does provide some thrust (very little for piston engines).

The amount of jet thrust for turboprops is in the range of 4%-15% of propeller-produced thrust. For a helicopter, it's only 2%-4% of rotor lift, but it's directed backwards. The helicopter in your picture would get 100-150 lbf of thrust from the engine exhaust, compared to 3,000-5,000 lbf of lift provided by the rotor.

This thrust is not controlled in any way. It helps a little in forward flight, and is compensated for by cyclic pitch in hover or maneuvering.

Residual thrust can be calculated, but is rarely specified in most sources. You can find some analyses of residual thrust for common engines online, such as for the PW-100 series, or for the Allison T-56.

The T-56's residual thrust, per above analysis, is 750 lbf, with 3442 kW on the shaft. That allows the T-56 engine to produce up to 10,000 lbf of thrust when used as a turboprop with a propeller. If used in a helicopter, such an engine would be able to deliver 25,000-40,000 lbf of lift with a rotor.