because our seasons are caused by..?
I doubt "seasons" would be defined by the Earth's tilt at that point. It wouldn't compare to the temperature change caused by moving between .3 AU and 1.7 AU away from the sun...
I'm aware your question regards Saturn but one thing I use to keep hearing is that 'Jupiter was nearly a star' which was always as quickly dismissed (this idea was even played on in 2010). If I'm not mistaken, Jupiter would have had to have been at least 10 times its current mass to have become a brown dwarf (though its radius would have remained approximately the same). The difference between brown dwarfs and large gas planets is that gas planets have a metallic core and the outer layers of gas remain seperate; with a brown dwarf, the metallic core mixes with the various gas layers through convection, material rising to the surface, cooling, and returning to the centre (on a side note, fusion in lower mass brown dwarfs doesn't necessarily take place, brown dwarfs bigger than 13 Jupiter masses fuse deuterium and those bigger than 65 Jm fuse deuterium and lithium). In order for either Saturn or Jupiter to have become brown dwarf stars, there might have been a need for at least 10-15 times as much material orbiting the sun as the solar system formed, I imagine this would have had some effect on the formation of the other planets.
Whoa hold on there for a sec, steve. Even if Jupiter were a brown dwarf Europa and Ganymede wouldn't produce no more of an atmosphere than they already have. For Europa and Ganymede is nothing more than chunks of ice and if they were to melt it would just be another complete seaworld in space.
It wasn't a flaw. The monoliths weren't eating the planet, they were adding mass to it by duplicating themselves from an extradimensional source - the monoliths are stargates to elsewhere. The extra mass caused it to begin collapsing, which is what we saw.
According to data obtained from the Galileo mission, Ganymede consists of a differentiated structure- a large lunar sized 'core' of rock & iron overlain by a deep layer of warm soft ice capped by a thin cold rigid water ice rich crust. If the surface temperature was to rise and the ice begin to melt, rather than there being rocky formations, Ganymede would probably become a sea world (possibly with the polar caps still frozen, depending on how much the temperature was to rise). As the core is estimated to be moon size, this would put it well below the surface of the new ocean. The following links should be of some use-
The Earth wouldn't orbit that barycenter. It would get tugged around with the Sun. Not much would change with our solar orbit. We would receive about 1/1000 the amount of energy from Saturn-star as we do from the Sun, so not much would change to Earth's seasons.