  Context. GJ 436b is one of the few transiting warm Neptunes for which a
detailed characterisation of the atmosphere is possible, whereas its
non-negligible orbital eccentricity calls for further investigation.
Independent analyses of several individual datasets obtained with Spitzer have
led to contradicting results attributed to the different techniques used to
treat the instrumental effects. Aims. We aim at investigating these previous
controversial results and developing our knowledge of the system based on the
full Spitzer photometry dataset combined with new Doppler measurements obtained
with the HARPS spectrograph. We also want to search for additional planets.
Methods. We optimise aperture photometry techniques and the photometric
deconvolution algorithm DECPHOT to improve the data reduction of the Spitzer
photometry spanning wavelengths from 3-24 {\mu}m. Adding the high precision
HARPS radial velocity data, we undertake a Bayesian global analysis of the
system considering both instrumental and stellar effects on the flux variation.
Results. We present a refined radius estimate of RP=4.10 +/- 0.16 R_Earth, mass
MP=25.4 +/- 2.1 M_Earth and eccentricity e= 0.162 +/- 0.004 for GJ 436b. Our
measured transit depths remain constant in time and wavelength, in disagreement
with the results of previous studies. In addition, we find that the
post-occultation flare-like structure at 3.6 {\mu}m that led to divergent
results on the occultation depth measurement is spurious. We obtain occultation
depths at 3.6, 5.8, and 8.0 {\mu}m that are shallower than in previous works,
in particular at 3.6 {\mu}m. However, these depths still appear consistent with
a metal-rich atmosphere depleted in methane and enhanced in CO/CO2, although
perhaps less than previously thought. We find no evidence for a potential
planetary companion, stellar activity, nor for a stellar spin-orbit
misalignment. [ABRIDGED]
