We present results from the high-precision timing analysis of the pulsar-white dwarf (WD) binary PSR J1012+5307 using 15 years of multitelescope data. Observations were performed regularly by the European Pulsar Timing Array (EPTA) network, consisting of Effelsberg, Jodrell Bank, Westerbork and Nançay. All the timing parameters have been improved from the previously published values, most by an order of magnitude. In addition, a parallax measurement of π= 1.2(3) mas is obtained for the first time for PSR J1012+5307, being consistent with the optical estimation from the WD companion. Combining improved 3D velocity information and models for the Galactic potential, the complete evolutionary Galactic path of the system is obtained. A new intrinsic eccentricity upper limit of e < 8.4 × 10−7 is acquired, one of the smallest calculated for a binary system and a measurement of the variation of the projected semimajor axis also constrains the system's orbital orientation for the first time. It is shown that PSR J1012+5307 is an ideal laboratory for testing alternative theories of gravity. The measurement of the change of the orbital period of the system of is used to set an upper limit on the dipole gravitational wave emission that is valid for a wide class of alternative theories of gravity. Moreover, it is shown that in combination with other binary pulsars PSR J1012+5307 is an ideal system to provide self-consistent, generic limits, based only on millisecond pulsar data, for the dipole radiation and the variation of the gravitational constant .