On constancy of cometary orbits how are the comets able to keep to a nearly fixed orbital period, though they lose a certain amount of mass during their perihelion?
 A: As seen from Kepler's laws the orbit is not dependent of the orbiting object's mass.  Also, if you do derivation for the orbit (usually in polar coordinates), coordinates $\theta$ and $r$, are orbiting object's mass independent:
$$ \ddot{r} - r \dot{\theta}^2 = -\frac{GM}{r^2}$$
$$ r \ddot{\theta} + 2 \dot{r} \dot{\theta}$$
If you are looking for more intuitive answer, the conclusions above are true due to the fact that gravitational force is dependent on the mass of the object $F_g \propto m$.  Since according to second Newton's law force is proportional to the mass of the object and its acceleration $F = m a$, when you want to find acceleration, the mass cancels out.  So as long as you have object's initial velocity, you can predict the orbit.
This is true, as dmckee pointed out, if the other object's (i.e. sun's) position is fixed.  In the case $m \ll M$, this is practically satisfied.
A: What exactly do you mean by a "nearly fixed orbital period"? For most comets the deviations from an orbit calculated based solely on gravitational parameters are on the order of fractions of a day per apparition, but for Comet 1P/Halley it is about four days. In any case, these deviations are well observed.
The paper Cometary Orbit Determination and Nongravitational Forces (D. K. Yeomans, P. W. Chodas, G. Sitarski, S. Szutowicz, and M. Królikowska) provides a nice overview (including some history). From the above linked:

[Friedrich] Bessel (1836) noted that
  a comet expelling material in a radial sunward direction
  would suffer a recoil force, and if the expulsion of material did not take place symmetrically with respect to perihelion, there would be a shortening or lengthening of the
  comet’s period depending on whether the comet expelled
  more material before or after perihelion ...
  Although Bessel did not identify the physical mechanism
  with water vaporization from the nucleus, his basic concept
  of cometary nongravitational forces would ultimately prove
  to be correct.

Also from the above paper:

The breakthrough work that allowed a proper modeling
  of the nongravitational effects on comets came with Whipple’s introduction of his icy conglomerate model for the
  cometary nucleus (Whipple, 1950, 1951). Part of his motivation for this model was to explain the so-called nongravitational accelerations that were evident in the motion of
  Comet Encke and many other active periodic comets. That
  is, even after all the gravitational perturbations of the planets were taken into account, the observations of many active comets could not be well represented without the introduction of additional so-called nongravitational effects into
  the dynamical model. These effects are brought about by
  cometary activity when the sublimating ices transfer momentum to the nucleus.

The paper goes on to detail the development of models and also summarizes a variety of observational studies over the last half century, and makes for an interesting read on the subject. 
