It's not entirely clear to me what you are asking. You seem to be asking several things at once, and your request for 'a thorough explanation of the influence of mach number on shockwaves, and how a shockless transition from supersonic to subsonic can be achieved' seems very broad. For instance, there are whole text books dealing with supersonic flow and the nature and behavior of shocks.
However, I will try to answer your questions as best I can, to see if I can help clear up any misconceptions:
isn't there always a shock wave present as long as the wave is supersonic?
I assume that here you are referring to a pressure wave travelling through a stationary gas? I believe, if the wave is travelling faster than the upstream speed of sound, then it must be a shock, since no other type of pressure wave can travel faster than sound. Note that it is only supersonic from the point-of-view of the upstream gas, which is flowing into the shock. The shock is moving subsonically, relative to the downstream gas (i.e. the gas that is leaving the shock). This is because the local density increases across the shock, so the local speed of sound downstream from the shock is higher than the shock velocity.
Does simply the type of shock (e.g. normal, oblique) change as alluded to here?
I am a bit unclear as to what exactly you are asking about here - does the type change due to what conditions? You seem to be referring to the inlet cone of a supersonic jet engine. In this case, the goal is to decelerate the flow from supersonic to subsonic (you want subsonic conditions for combustion in most typical engines) with minimal loss of energy. This is done by firstly sending the flow through a series of oblique shocks and then finally through a (weak) normal shock. This is good because oblique shocks tend to be weaker than normal shocks and incur smaller losses. If there was no inlet cone, then the supersonic flow would have to go through a huge normal shock in front of the engine, which would be very lossy and inefficient.
Is there any difference from M=1 shock waves and M>1 shock waves?
I think you may have a misconception here. Shock waves are supersonic flow phenomena, so technically you can't have a shock at M=1. At M=1, you are just on the verge of causing a shock, but you won't actually have a shock until you exceed the sound barrier. Sorry if that seems a bit pedantic, but I think it's an important point: there are no 'M=1' shocks, only M>1 shocks. If the flow is at M=1.00000001, then it's still M>1. (note that here I am referring to the local Mach number of the flow, not the global 'average' Mach number, e.g. for an aircraft)
The key difference to understand is between normal and oblique shocks. Normal shocks are stronger and always go from M>1 to M<1, i.e. they take flow from supersonic to subsonic. Oblique shocks are less strong. They can take flow from M>1 to M<1, but they can also go from M>1 to M>1, i.e. from supersonic to supersonic (but lower M). In fact, in the direction normal to the shock, an oblique shock is also taking flow from supersonic to subsonic, however the flow downstream of the oblique shock is still supersonic, because there is a component of flow velocity tangential to the shock, which is not altered by the shock. This Wikipedia page would be a good place to start, if you want to learn more about oblique shocks.
For example, this article describes a shockless transition from supersonic to subsonic flow, but what does that conceptually mean?
I didn't read the whole article, but it looks like he is referring to slowing down a supersonic flow at the inlet of a supersonic jet engine, as I mentioned above, but in a way that avoids having any shocks, which would be ideal, because then it would avoid those losses. This seems plausible, because if you look at 1D compressible flow theory, flow velocity responds to changes in the flow passage area - supersonic flow should slow down if the flow area is reduced. So, it seems like he may have found some clever geometrical technique that enables the flow to be slowed down to subsonic without having to go through those entropy-generating shocks (which I am sure would be of interest in supersonic engine design).
I hope that helps.
Edit:
For 1D isentropic flow, this is the key equation that relates duct area to velocity:
$$\frac{dV}{dA}=\frac{V}{A(M^2-1)}$$
where $V$ is the velocity, $A$ is the duct area and $M$ is the Mach number. You can see from this that if $M<1$, $\frac{dV}{dA}$ is negative and if $M>1$, $\frac{dV}{dA}$ is positive. Thus, it is possible in theory to decelerate a supersonic flow isentropically by reducing the flow area, without needing any shocks.
If you want to know more about this, then I think reading through a few chapters of a compressible flow text book would help you a lot and would help answer many of your questions.
