The Parkes multibeam pulsar survey: VI. Discovery and timing of 142 pulsars and a Galactic population analysis

[ABRIDGED] We present the discovery and follow-up observations of 142 pulsars found in the Parkes 20-cm multibeam pulsar survey of the Galactic plane. These new discoveries bring the total number of pulsars found by the survey to 742. In addition to tabulating spin and astrometric parameters, along with pulse width and flux density information, we present orbital characteristics for 13 binary pulsars which form part of the new sample. Combining these results from another recent Parkes multibeam survey at high Galactic latitudes, we have a sample of 1008 normal pulsars which we use to carry out a determination of their Galactic distribution and birth rate. We infer a total Galactic population of 30000 +/- 1100 potentially detectable pulsars (i.e. those beaming towards us) having 1.4-GHz luminosities above 0.1 mJy kpc squared. Using a pulsar current analysis, we derive the birth rate of this population to be 1.4 +/- 0.2 pulsars per century. An important conclusion from our work is that the inferred radial density function of pulsars depends strongly on the assumed distribution of free electrons in the Galaxy. As a result, any analyses using the most recent electron model of Cordes & Lazio predict a dearth of pulsars in the inner Galaxy. We show that this model can also bias the inferred pulsar scale height with respect to the Galactic plane. Combining our results with other Parkes multibeam surveys we find that the population is best described by an exponential distribution with a scale height of 330 pc.