If a "relativistic mass" depends on direction is a matter of its definition. E/c² for example is independent from direction. There is no reason to start such discussions over and over again. It is sufficient to advice against the use of relativistic mass. That has been done in this thread.
That's why it is obviously something completely different than mass - no matter whether you discuss the concept within Newtonian or relativistic physics.
The non-relativistic limit of the relativistic mass also coincides with the mass in Newtonian mechanics and it is even additive. It doesn't...
How can we see that if we don't even know what "amount of material" means? I looked for a definition but I found nothing that would make sense in this context. Only @pervect can clarify what he is talking about. But if it turnes out to be something related to Newtonean mechanics, he (and not me)...
Of course a frame dependent property cannot be interpreted as a relativistically invariant measure. But if we cannot discuss what "amount of material" is and wheather is must be relativistically invariant or not this will get us to nowhere. Better keep it out of the discussion.
That depends on the meaning of "mass in the sense of Newtonian physics". But you can't explain that without talking about Newtonian mechanics and we have been instructed not to do that in this thread. Thus, this doesn't seem to be the right place for such statements.
That depends on the context. Newton's term for mass was "quantitas materiae".
I do not think it is that easy. For the case that the box is at rest I get
##m_{system}^2 = \left( {m + M + \frac{{E_{kin} }}{{c^2 }}} \right)^2 - \left( {2 \cdot m + \frac{{E_{kin} }}{{c^2 }}} \right) \cdot...
That depends on how you do it and what you start with. But we don't need to discuss that in detail. It is sufficent to say that relativistic mass is not useful and should be avoided.
Of course it means something but it is outdated. It is just about OK in special relativity but using it in general relativity would be like going with horse and knight's armour into a modern battle.
That would be a very long list and it would never be complete. But typical reactions are usually listed in sources for the individual compounds (e.g. chemistry textbooks or Wikipedia). For example the French Wikipedia entry for NaH includes the above mentioned reaction with HCl.