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Aperture synthesis observations of HCO+ J = 1-0, 13CO 1-0, and C18O 1-0 obtained with the Owens Valley Millimeter Array are used to probe the small-scale (5" approximately 700 AU) structure of the molecular envelopes of a well-defined sample of nine embedded low-mass young stellar objects in Taurus. The interferometer results can be understood in terms of: (1) a core of radius approximately or less than 1000 AU surrounding the central star, possibly flattened and rotating; (2) condensations scattered throughout the envelope that may be left over from the inhomogeneous structure of the original cloud core or that may have grown during collapse; and (3) material within the outflow or along the walls of the outflow cavity. Masses of the central cores are 0.001-0.1 M (solar), and agree well with dust continuum measurements. Averaged over the central 20" (3000 AU) region, an HCO+ abundance of 4 x 10(-8) is inferred, with a spread of a factor of 3 between the different sources. Reanalysis of previously presented single-dish data yields an HCO+ abundance of (5.0 +/- 1.7) x 10(-9), which may indicate an average increase by a factor of a few on the smaller scales sampled by the interferometer. Part of this apparent abundance variation could be explained by contributions from extended cloud emission to the single-dish C18O lines, and uncertainties in the assumed excitation temperatures and opacities. The properties of the molecular envelopes and outflows are further investigated through single-dish observations of 12CO J = 6-5, 4-3, and 3-2, 13CO 6-5 and 3-2, and C18O 3-2 and 2-1, obtained with the James Clerk Maxwell and IRAM 30 m telescopes, along with the Caltech Submillimeter Observatory. Ratios of the mid-J CO lines are used to estimate the excitation temperature, with values of 25-80 K derived for the gas near line centre. The outflow wings show a similar range, although Tex is enhanced by a factor of 2-3 in at least two sources. In contrast to the well-studied L1551 IRS 5 outflow, which extends over 10' (0.4 pc), seven of the remaining eight sources are found to drive 12CO 3-2 outflows over < or = 1' (0.04 pc); only L1527 IRS has a well-developed outflow of some 3'(0.12 pc). Estimates are obtained for the outflow kinetic luminosity, Lkin, and the flow momentum rate, FCO, applying corrections for line opacity and source inclination. The flow force FCO correlates with the envelope mass and with the 2.7 mm flux of the circumstellar disk. Only a weak correlation is seen with Lbol, while none is found with the relative age of the object as measured by integral Tmb(HCO+ 3-2)dV/Lbol. These trends support the hypothesis that outflows are driven by accretion through a disk, with a global mass infall rate determined by the mass and density of the envelope. The association of compact HCO+ emission with the walls of the outflow cavities indicates that outflows in turn influence the appearance of the envelopes. It is not yet clear, however, whether they are actively involved in sweeping up envelope material, or merely provide a low-opacity pathway for heating radiation to reach into the envelope.