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I think the problem is that while in the first case your differential equation applies to all your domain of interest and you can just use it, in the second situation the DE doesn't apply at x=0. This means you need to solve the DE at the domains $0<x<L/2$ and $-L/2<x<0$ separately, where it does still apply. When applying it to those domains ...


You're on the right track here. For convection between plates to 'stop' (it won't really stop, just get wicked slow) you want viscosity to be very high relative to buoyancy, so you want low Grashof. Also, in the case of very narrow plates, you'd want to use the gap width as your characteristic length $L$. $10^8$ is the transition point at which the flow ...

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