The Camera is called 'Tianhe Inner Gimbal camera B'
It is a CCTV dome, but it was stowed/uninstalled when first seen.
(This is a partial answer, as it doesn't answer the ISS part)
Seen here before launch:
After the crew enter:
So between arrival on 17 June, and before President Xi Jinping's address to them, the crew do a variety of work, some of which is ...
If it's not "a solid or liquid propellant", and more fuel-efficient, and not magical, it's presumably some sort of electric thruster.
This Interesting Engineering article (CW: journalism) says Tiangong uses ion thrusters:
The space station’s core Tianhe module, which will welcome its first astronauts later this month if all goes to plan, is ...
The characters in the flag appear to be "搜救 回收大队" which Google translate gives as: "Search and rescue recovery brigade"
Though I haven't found a direct link to an organization with that flag, all the references point to it being the search and rescue unit of the Jiuquan Satellite Launch Center
This article describing the ...
tl;dr: It could be dozens of years from now or it could be next week!
Since the artificial satellite orbits are constantly changing, what's necessary is to regularly run a conjunction search program that updates itself with the latest TLEs for the satellites, and uses them to look far enough ahead that you can book a flight.
As your map tells us, two ...
Partial answer to
Is there a counterpart on the ISS and if so what does it look like?
Using the working hypothesis that the "dark, ceiling-mounted hemisphere" is a ground-controlled camera, then no, there is no counterpart on the US side of the ISS.
The answer to this question How closely can ground control monitor ISS astronauts? discusses the ...
A bit dated (2011) article from China Daily states that the launch of Tiangong-3 is planned around 2020-2022, stating that this is information from The China Manned Space Engineering Office. The guardian had an article from around the same time, giving a similar time range. A more recent article from Space Daily states that the plan is not changed.