Does a right circular cone only consists of pair of straight lines, hyperbolas, parabola, circles and ellipses? I was reading about the conic sections and that a conic section includes pair of straight lines, ellipses, hyperbola, circles and parabola.
Are all these 5 components enough to form a right circular cone or are these just the parts of right circular cone? What I mean is : if I put together infinite number of circles, parabolas, hyperbolas,, ellipses and circles, will I get a right circular cone?

Yes, a right circular cone consists only of pair of straight lines, hyperbolas, parabola, circles and ellipses when in each case it is cut transverse planes.
Take a cone of a fixed semi-vertical angle $\alpha$ whose symmetry axis is the x-axis 
Case 1
Cut it by a plane of variable inclination $ \beta$ and constant $C$. 
$$y = \tan \beta \,x + C \tag{1} $$  
The conic section so formed has the eccentricity
$$ \epsilon = \dfrac{\sin\alpha }{\sin \beta} \tag{2}$$
CASE  $ c \ne 0 $ i.e., cutting plane avoids the cone apex:
If $\beta > \alpha $ you get an ellipse.
If $\beta < \alpha $ you get a hyperbola in two disjointed parts.
If $\beta = \alpha $ you get a parabola. 
If you rotate any of the three above conics about the x-axis you get back the cone cut segment.
CASE  $ c = 0 $ i.e., cutting plane includes the cone apex:
A pair of straight lines which include an angle $ 2 \gamma $
$$ \tan \gamma =  \tan \alpha \,\cos \beta \tag{3}$$ 
Case 2
You can keep $\beta $ constant and $C$ variable, you get back the cone of stacked conic sections 
If you dilate and shift any conic by multiplication  factors $ (\sin \alpha, \cos \alpha )$ respectively along x-axis 
This response would be more convincing with images. 
Using expression (4) uploading a (poor quality) image of a two dependent parameter $ r,\theta $ cone surface where one of $ \beta, C $ is varied at a time, keeping the other fixed. Ellipses are stacked to make up the cone.All type three types of conics are seen on one cone sheet. 
(Cone apex is excluded in the first plot. The parabola is not plotted in the second .. due to limitations in expression when some quantities go infinite).

Parametric Equations of all possible conics
$$ (x,y,z) = (x, \tan \beta\, x - c , \pm \sqrt {x^2( \tan{^2} \alpha  -\tan{^2} \beta) + 2 \tan \beta\, c \,x - c^2} ) \tag{4} $$
The process by which you get back the cone from conic sections is in two ways.
Rotation through x-axis rotation, and, 
Dilation/Translation keeping conic on x-axis.
