Will the connection resistances influence the magnitude of the current? What instruments may be used for the purpose of measuring and recording microcurrents?
The resistance brought by connections can usually be counted as a series resistance. As a consequence, for a given **applied** current this does not affect the measure. **But if you apply a voltage** and try to measure the resultant current, the effect of contact resistances is to decrease the effective voltage on the device you try to measure (in other words, the effective voltage is not anymore the voltage you rul on the supply). So depending on the device impedance / contact resistance ratio, it can affect in different manners the current you measure.
That said, microampers are not very low currents, and you should be able to easily measure them with classical ampermeters with sufficiently low resolution.
I agree with the last comment, it's important to know the frecuency and the capacity of tha cables. But if you try to measure a DC current it will be easy using regular BNC connectors and coaxial cables. The Keithley 6482 is a common choice.