I like to view the sign convention as, the positive direction is the direction the light is moving. (This means the incident and reflected light will have opposite positive directions.)
Example: An object is on the left, and the concave mirror is on the right. They are relatively far away $(d_o>f)$. Light rays travel to the right from the object to the mirror. So, light traveling from the object to the mirror moves a positive amount, and $d_o$ is positive. After the light reflects it is moving back to the left, so left is the positive direction for the reflected light. For this example, the image will be formed to the left of the mirror, and since left is positive, $d_i$ will be positive.
Another example: An object on the left, and concave mirror on the right, just as before, but now they are very close $(d_o<f)$. The light traveling from the object to the mirror moves right, so right is positive, and $d_o$ is positive, as before. The reflected light travels back to the left, making left the positive direction for reflected light, also as before. However, in this example, the reflected rays diverge. If you use geometric ray tracing on this example, you will trace the rays backwards behind (to the right of) the mirror, until they converge. But since left is positive for the reflected light, the image to the right will have a negative $d_i$.
This same idea helps to explain why a convex mirror has a negative focal length. Parallel light will reflect and diverge, and tracing the reflected rays shows them converging to the behind (to the right of) the mirror, which is in the negative direction for reflected light.
You are getting reflections from the front (glass surface) and back (mirrored) surface, including (multiple) internal reflections:
It should be obvious from this diagram that the spots will be further apart as you move to a more glancing angle of incidence. Depending on the polarization of the laser pointer, there is an angle (the Brewster angle) where you can make the front (glass) surface reflection disappear completely. This takes some experimenting.
The exact details of the intensity as a function of angle of incidence are described by the Fresnel Equations. From that Wikipedia article, here is a diagram showing how the intensity of the (front) reflection changes with angle of incidence and polarization:
This effect is independent of wavelength (except inasmuch as the refractive index is a weak function of wavelength... So different colors of light will have a slightly different Brewster angle); the only way in which laser light is different from "ordinary" light in this case is the fact that laser light is typically linearly polarized, so that the reflection coefficient for a particular angle can be changed simply by rotating the laser pointer.
As Rainer P pointed out in a comment, if there is a coefficient of reflection $c$ at the front face, then $(1-c)$ of the intensity makes it to the back; and if the coefficient of reflection at the inside of the glass/air interface is $r$, then the successive reflected beams will have intensities that decrease geometrically:
$$c, (1-c)(1-r), (1-c)(1-r)r, (1-c)(1-r)r^2, (1-c)(1-r)r^3, ...$$
Of course the reciprocity theorem tells us that when we reverse the direction of a beam, we get the same reflectivity, so $r=c$ . This means the above can be simplified; but I left it in this form to show better what interactions the rays undergo. The above also assumes perfect reflection at the silvered (back) face: it should be easy to see how you could add that term...
Best Answer
I thought that this was an interesting question but I am afraid I have no definitive answer.
The standard concave mirror formula where the radius of curvature is twice the focal length with the linear magnification $m = \dfrac{\text{image size}}{\text{object size}} = \dfrac{\text{image distance }}{\text{object size}}$ should enable one to decide what a magnification of $5\,\rm x$ means but it seems to be not quite as easy as that.
What is certain is that you the object)) must place your face at a distance less than the focal length of the mirror to produce a virtual, upright and magnified mirror of your face in the mirror.
Also that image must be further than about $25\, \rm cm$ from your eyes (least distance of distinct vision$. otherwise it will be out of focus.
Judging by the variety of specifications for a $\rm 5x$ mirror either there is no regulation or nobody adheres to the regulation.
These mirror seem to be made mostly in China and sometimes as well as the magnification eg $\rm 5\,x$ a radius of curvature for the concave mirror is given as $\rm 600R$, meaning $\rm600\,mm$.
One website quotes $\rm 3x, 700R; 5x, 600R; 7x, 560R;$ and $\rm 10x, 535R$ whereas another website states $\rm 2x, 100R; 3x, 600R; 5x, 400R;$ and $\rm 7x, 300R$.
They also like to quote plane mirrors as $\rm 1\,x$, so magnification is $\dfrac{\text{image size}}{\text{object distance}}$.
Another website assumes that the object distance is $240 \,\rm mm$ so it is likely that the magnification is related to the object (you) being $240 \,\rm mm$ from a concave mirror and comparing the size of the image as seen in a concave mirror with size as seen in a plane mirror when you are $240 \,\rm mm$ from a mirror.
This works out for the first website as with a radius of curvature of $\rm600\,mm$ , which is a focal length of $\rm 300\,mm$ and an object distance of $\rm 240\,mm$ the image distance comes out to be $\rm 1200\,mm$ giving a magnification of $\rm 5x$.
For the second website to get a magnification of $\rm 5\, x$ with a concave mirror which has a radius of curvature of $\rm 400\,mm$ requires the object distance to be $\rm 130\,mm$.
Perhaps the object distances are related to the least distance of distinct vision which is usually set at $\rm 250\,mm$ for a "average" eye?
The second website has then reasoned that with the average eye putting a mirror $\dfrac {25}{2}\, \rm mm$ away from you, the image of yourself which you see in the mirror will be $\rm 250\,mm$ away from you and still in focus.
This would give the largest possible image (visual angle) and hence this should be the reference object distance.
It might help you to get an answer if you have access to such mirrors to measure their radius of curvature with the method of no parallax probably the easiest way of doing it.
Lay the mirror on the floor.
Look down at it from above and move your head around until you see an image of your eye.
Hold a pin (a finger will do) between your eye and the mirror and move it until it is covering the image of your eye.
At the same time the image of the pin will appear.
Move the pin up and down until the pin and its image are the same size and do not move relative to one another as you move your head from side to side - a position of no-parallax between the pin and its image.
The distance between the pin and the mirror is the radius of curvature of the mirror.
If you accumulate some data it might determine if there is some semblance of order amongst the manufactures of these mirrors.
Unfortunately I only have a $2000\rm R$ (measured using no parallax) mirror in the house and it is not labelled as to what its magnification is.