@Marko7 let me try to answer some of this
1)
As the water is flowing through the core pipes from bottom to top with the help of the circulation pumps it gets heated up , this adds to the flow so at normal operation the water starts to form steam in it (steam voids) the further up you go along a pipe , so at the top where the water exits each pipe it has some percentage of steam in it. Then it moves through pipes to the steam separator, as it gets in there the fraction that is liquid falls on the bottom as it's heavier while the lighter steam void fraction tends to rise up because it's lighter , some of that steam condenses in the separator falling back down and joining the water at the bottom , the rest goes out to the turbine hall where it enters the steam turbine.2)
Although I don't have direct links but I would presume that the gas loop is closed and the gas is recirculated from the reactor core cavity through filters and back again, unwanted gasses that have been created due to the operation of the core are filtered out from this loop and stored at a separate chamber for some days until their short lived radioactivity drops to a safe level and then are allowed to exit through the chimney, that is the main purpose of the stack seen above RBMK units.
Now here is another interesting feature and maybe the reason why the RBMK stack is so big and high, from what I read it is also made as an emergency steam dump in case the external reactor pipework bursts, namely the endless pipes going in and out of the steam separators , so that there doesn't form dangerous overpressure inside the building. Maybe
@Astronuc can say something about this, at least this is what I read in the Russian language papers regarding the RBMK.
I think that in the PDF you linked in your post #16 they were considering this method as one of the possible emergency passive core cooling ways, because the RBMK has a lot of individual pipes and the steam separators themselves are very large having a large surface area essentially forming a natural radiator/heat exchanger so they though of using water sprays on to the steam separators to cool them down and let the vapor exit the original pathways to the chimney?3)
at least in RBMK diesels are solely to generate electricity then used to power equipment like circulation pumps etc. as far as I know all NPP have the same way and I haven't heard of diesels powering pump directly.5)
This seemingly simple question is actually complicated but in short, output thermal power or heat capacity is increased when the number of nuclear reactions/neutrons in the reactor within given time period is increased.
Insert more rods = more neutrons get absorbed. more neutrons get absorbed = less fission reactions in one second, core cools down.
This is the reactivity part but coolant temp also has a relationship which actually played a crucial role in the 1986 accident. You see water is also a neutron moderator/absorber so as you get less water in the channels you lose some percentage of neutron absorbing capacity of the core.
The higher the power of the core the hotter the water becomes for any given water flowrate, the more steam is in the water it becomes less dense and absorbs less neutrons which then increase the reactivity.
Here is a catch , graphite is a better, more economical neutron moderator as it reflects more of them and absorbs less than water, so as you decrease water percentage in the core pipes more and more moderation is done by the core graphite so the neutron flux increases , this is the positive void coefficient of the RBMK.7)
For this info i think you would have to search the internet in Russian as most of the data with regards to RBMK is in Russian, unlike the VVER Soviet made reactors the RBMK were not exported outside the USSR so they are solely a Soviet/Russian thing and most of their details are only known by Russian specialists.
How good are you with Russian language ?