  We present results from the high precision timing analysis of the
pulsar-white dwarf (WD) binary PSR J1012+5307 using 15 years of multi-telescope
data. Observations were performed regularly by the European Pulsar Timing Array
(EPTA) network, consisting of Effelsberg, Jodrell Bank, Westerbork and
Nan\c{c}ay. All the timing parameters have been improved from the previously
published values, most by an order of magnitude. In addition, a parallax
measurement of $\pi = 1.2(3)$ 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 $e<8.4\times 10^{-7}$ is acquired,
one of the smallest calculated for a binary system and a measurement of the
variation of the projected semi-major 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 $\dot{P}_{b} = 5(1)\times
10^{-14}$ 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 $\dot{G}$.
