I am honestly confused about your question, but no worries, I will try to be as thorough as possible in the hope of helping you. At this point, I will not prove any equation from first principles (Newtow's Laws), but should you want that or any further clarification, just ask.
First let´s have a look at the equation of motion:
$$\sum \vec{M_{O_i}^{ext}} = \dot{\vec{H_O}} $$
In this equation the net external moment (also known as net external torque), $$\sum \vec{r_{i/O}} \times \vec{F_i^{ext}}$$, is equated to the time derivative of angular momentum. The assumption of central internal forces was made to avoid having to sum internal moments as well, but that's nothing to worry about since in typical mechanical systems that assumption is valid. It should be noted that point O has to either be a fixed point in space or the center of gravity (G).
$\vec{H_G}$ can be calculated using the following equation. The moments and products of inertia should be calculated with respect to point G.
$$\vec{H_G} =
\begin{bmatrix}
I_{xx} & -I_{xy} & -I_{xz} \\
-I_{yx} & I_{yy} & -I_{yz} \\
-I_{zx} & -I_{zy} & I_{zz}
\end{bmatrix}
\cdot
\begin{bmatrix}
\omega_x \\
\omega_y \\
\omega_z
\end{bmatrix}
$$
For any other point in space,
$$
\vec{H_O} = \vec{H_G} + \vec{r_{G/O}} \times \vec{L}
$$
, where $\vec{L}$ is the linear momentum (which is equal to $M\cdot\vec{v_G})$
Note: Remember to take the time derivative of the angular momentum when using the equation of motion.
Now that the fundamentals have been established, I will address some specific points in your question.
(i) angular momentum of a rigid body has a parallel and perpendicular component to the angular velocity. True. Unless the angular velocity is an eigenvector (of the inertia matrix), the angular momentum does not point in the direction of the angular velocity.
(ii) perpendicular component is zero if the axis of rotation is also symmetry axis of rigid body. Rotational symmetry about an axis, means that if the inertia tensor is rotated until that axis matches one of the coordinated axis (x,y,z), the inertia tensor will then be diagnolized and two of its moments of inertia will be the same. In such a case, an angular velocity pointing in the direction of the axis of symmetry is clearly an eigenvector, so there are no perpendicular components.
(iii) why the angular momentum with respect to the center of mass can also has a perpendicular component. That depends on the angular velocity, if it is an eigenvector, angular momentum will point in the same direction. Otherwise, it may have other components.
If you are struggling to accept what the equations are showing you, I suggest you go through the trouble of deriving yourself the equations.
The last thing I would like to mention is your comments on "constraints". The constraits in problems typically are either connected with the loading (forces and moments) or with the kinematics (such as the direction of the angular velocity or the velocity of a particular point), NOT with the direction of the angular momentum, much less, the direction of its derivative.
