I am trying to get a common understanding from these two previous questions:
Does the increase of mass occur only if the particle approaches c (speed of light in a vacuum) or if it simply approaches the speed of light in its current medium? For example, does the mass of charged particles increase during Cherenkov radiation?
First, your question.
Yes, the increase of mass occurs only when a particle approaches $c$ (speed of light in vacuum).
$c$ is fundamental in Special Relativity, not because it is the speed of photons, but because it is the constant speed in the universe (the only speed invariant to boosts). Just because macroscopic light is transmitted at a lower speed inside a particular medium, that doesn't mean that the fundamental speed of Special Relativity is any different. Even inside mediums where light travels more slowly, all relativistic effects happen when a particle approaches $c$.
Since Cherenkov radiation (CR) is just an effect related to the speed of light in a medium (and not to $c$), it doesn't have anything to do with mass increase either. Though CR and mass increase can happen simultaneously to a particle, they are independent (the first does not imply the second, and vice-versa).
Second, about the increase of mass.
It has been a historical habit to say that a particle's mass increases as $m=\gamma m_0$ when its velocity approaches $c$. That is not very appropriate. While it may seem convenient to define this relativistic mass, it's not a good habit.
First, because it's confusing to some people. There are physically intuitive ways to explain to a student why time intervals must stretch and why space intervals have to contract, but there's no way at all to explain why a particle's mass should increase.
Second, it's also not accurate. The defined relativistic mass parameter does not sustain the properties you would expect from a mass under close analysis. (I have a reference for this, but the pdf file somehow got corrupted in the last 8 years. I'm looking for a copy.)
Special relativity is based on the concept of a single absolute "speed limit" for the universe, from all (inertial) reference frames. That "speed limit" is the constant $c$, which is equivalent to the speed of light in a vacuum. The speed of light in other materials is not fundamental in any way.
So the simple answer is, the two points you bring up are quite orthogonal. Sure, a particle undergoing Cherenkov radiation is likely to be going a high percentage of the speed of the light, and would thus have increased mass. (Note that the Cherenkov radiation would decrease its energy and thus relativistic mass somewhat, due to the EM interaction with the surrounding molecules of the material). At the end of the day, consider 1. as a relativistic effect and 2. as a electromagnetic one. They're both related of course, but there's no direct connection that I think you're looking for.
the mass of an object increases closer to the speed of light because you can not pass the speed of light and the more energy you pump in, it will be converted to mass as the equation predicts "E=MC2" and this the cause for time dilation, because gravity is inversely proportional to time.
