I can't picture what you're thinking. A star in space is a great example of an OPEN system, as energy is copiously pouring across any nearby closed boundary.
Not even close.
Well, it is an idealization of the sun as an infinite black body at its surface temperature, so that the surface serves as a "pinhole". This is a very nice theoretical model of the sun for many purposes, so I say it is "close" in that sense.
In a more physically accurate model, the interior of the sphere would become filled with black body radiation at the sun's surface temperature very quickly ( hours or days, I think ) and then would slowly heat up as the sun heated up.
The absorption destroys entropy. The reemission is what generates entropy.
I don't think so. Cf. the 1854 definition of entropy. When heat Q flows from a higher temperature T1 to a lower temperature T2 the change in entropy, Q/T2 - Q/T1, is positive.
I suppose you're thinking of the entropy reduction involved in the production of biological materials such as cellulose. Here, the absorption of radiation increases the entropy of the absorbing cell, and the dissipation of this energy lowers the entropy, at the same time "carrying away" the entropy absorbed from the constituents of the cellulose leaving them in the lower entropy "organized" structure. Of course, the net effect is an increase of entropy in the cell and surroundings, just as a refrigerator is a net entropy generator, even though it lowers the entropy of warm items introduced into it via cooling.
In the case of biological processes, the entropy absorbed in the creation of organized structures is ultimately carried away by the earth's radiation into space.
Of course, the amount of entropy absorbed in these life processes is less than miniscule in comparison with the entropy produced by the absorption and radiation of solar energy by gross physical processes.