The answer to your first question is (now) fairly simple: No, brown dwarfs are not more common than red dwarfs. A crude approximation is that stars (which are indeed mostly red dwarfs) outnumber brown dwarfs 4 or 5 to 1; see, for example, the review article by Chabrier et al. (2014). This is supported by extensive surveys done in the 2000s and 2010s, including observations with the WISE telescope; these are sensitive enough to uncover quite faint brown dwarfs.
Empirically, attempts to count stars and brown dwarfs in local volumes strongly suggest that stars outnumber brown dwarfs. For example, Reylé et al. (2021) combine Gaia observations (sensitive to hydrogen-burning stars and the more massive brown dwarfs) with literature observations of brown dwarfs using IR telescopes (Spitzer, WISE) to make a census of different objects within 10 pc (33 light years) of the Sun. They find a total of 355 (hydrogen-burning) stars -- including 276 M dwarfs --, 20 white dwarfs (formerly hydrogen-burning), and 85 brown dwarfs (plus 3 more candidates). They do note that there are probably more brown dwarfs waiting to be identified, though these would have to be at the very low-mass (and thus low-luminosity) end. One can still argue that stars outnumber brown dwarfs in our local volume by about three or four to one.
It is true that lower mass stars tend to be more common than higher mass stars. However, this trend becomes weaker for lower masses. The figure below (from the Wikipedia article on the initial mass function (IMF) of stars, which tells you how many stars of a given mass are born, as a function of mass) shows several different versions of the IMF. The best current measurements are encoded in either the "Kroupa01" or "Chabrier03" curves, all of which show that the curves flatten -- or even turn over -- for very low masses. This means that if you add up all the brown dwarfs and add up all the stars, you end up with more stars than brown dwarfs.
Edited to add: Empirical counts of stars and brown dwarfs in the local volume.