This article is from the Astronomy FAQ, by Joseph Lazio (firstname.lastname@example.org) with numerous contributions by others.
The various parts of this question will be considered separately.
Also, rather consider how many stars there are in the Universe, we'll
consider how many stars there are in the Milky Way. The number of
stars in the Universe can be estimated by multiplying the number of
stars in the Milky Way by the number of galaxies in the Universe.
Subject How many stars are there in the Milky Way?
By William Keel <email@example.com>
My standard answer in introductory astronomy classes is "about as many
as the number of hamburgers sold by McDonald's." Being more precise
requires an extrapolation, because we can't see all the individual
stars in the Milky Way for two reasons---distance and dust absorption.
Both factors make stars appear dimmer. Observations at visible
wavelengths are limited to a region of (more or less) 5000 light-years
radius about the Sun, with a few windows in the intervening dust
giving us glimpses of more distant areas (especially near the Galactic
center). Our map of the Galaxy gets correspondingly more sketchy with
distance. Guided somewhat by observations of other spiral galaxies, we
think that the overall run of star density with radius is fairly well
known. Getting a total stellar head count is more of a problem,
because the stars that we can see to the greatest distances are also
the rarest. Measurements of the relative numbers of stars with
different absolute brightness (known in the trade as the luminosity
function) shows that, for example, for every Sun-like star there are
about 200 faint red M dwarfs. These are so faint that the closest,
Proxima Centauri, despite being closer to the Sun than any other
(known) star, takes very large binoculars or a telescope to find. So,
to get the total stellar population in the Milky Way, we must take the
number of luminous stars that we can see at large distances and assume
that we know how many fainter stars go along with them. Recent numbers
give about 400,000,000,000 (400 billion) stars, but a 50% error either
way is quite plausible. Much of the interest in "brown dwarfs" stems
from a similar issue---a huge number of brown dwarfs would not change
how bright the Galaxy appears (at visible wavelengths), but would
change its total mass quite substantially. Oddly enough, within a
particular region, we probably know the total mass and luminosity
rather more accurately than we do just how many stars are producing
that light (since the most common stars are by far the dimmest).