Good effort for thinking through this issue. Here's an incomplete answer, but some starters for the next steps.
First, the AIC scores - based on likelihoods - are on different scales because of the different distributions and link functions, so aren't comparable. Your sum of squares and mean sum of squares have been calculated on the original scale and hence are on the same scale, so can be compared, although whether this is a good criterion for model selection is another question (it might be, or might not - search the cross validated archives on model selection for some good discussion of this).
For your more general question, a good way of focusing on the problem is to consider the difference between LOG.LM (your linear model with the response as log(y)); and LOG.GAUSS.GLM, the glm with the response as y and a log link function. In the first case the model you are fitting is:
$\log(y)=X\beta+\epsilon$;
and in the glm() case it is:
$ \log(y+\epsilon)=X\beta$
and in both cases $\epsilon$ is distributed $ \mathcal{N}(0,\sigma^2)$.
Although it may appear that the mean of the log-transformed variables is preferable (since this is how log-normal is typically parameterised), from a practical point of view, the log of the mean is typically much more useful.
This is particularly true when your model is not exactly correct, and to quote George Box: "All models are wrong, some are useful"
Suppose some quantity is log normally distributed, blood pressure say (I'm not a medic!), and we have two populations, men and women. One might hypothesise that the average blood pressure is higher in women than in men. This exactly corresponds to asking whether log of average blood pressure is higher in women than in men. It is not the same as asking whether the average of log blood pressure is higher in women that man.
Don't get confused by the text book parameterisation of a distribution - it doesn't have any "real" meaning. The log-normal distribution is parameterised by the mean of the log ($\mu_{\ln}$) because of mathematical convenience, but equally we could choose to parameterise it by its actual mean and variance
$\mu = e^{\mu_{\ln} + \sigma_{\ln}^2/2}$
$\sigma^2 = (e^{\sigma^2_{\ln}} -1)e^{2 \mu_{\ln} + \sigma_{\ln}^2}$
Obviously, doing so makes the algebra horribly complicated, but it still works and means the same thing.
Looking at the above formula, we can see an important difference between transforming the variables and transforming the mean. The log of the mean, $\ln(\mu)$, increases as $\sigma^2_{\ln}$ increases, while the mean of the log, $\mu_{\ln}$ doesn't.
This means that women could, on average, have higher blood pressure that men, even though the mean paramater of the log normal distribution ($\mu_{\ln}$) is the same, simply because the variance parameter is larger. This fact would get missed by a test that used log(Blood Pressure).
So far, we have assumed that blood pressure genuinly is log-normal. If the true distributions are not quite log normal, then transforming the data will (typically) make things even worse than above - since we won't quite know what our "mean" parameter actually means. I.e. we won't know those two equations for mean and variance I gave above are correct. Using those to transform back and forth will then introduce additional errors.
Best Answer
It's not true. It is not the case that $\text{Var}(\log(Y)|X=x)$ being constant implies $\text{Var}(Y|X=x)$ is constant:
-- in fact that's only the case if the mean is constant.
This problem appears to be caused by the original omitting to show the fact that there's conditioning on $x$ and then forgetting that it had done so.
Note, however, that the assumption of constant variance on the log scale is untrue. If you generate data from a Poisson regression model and take logs, the conditional variance is not constant. (and this is possibly what the text was trying to explain)
Taking logs makes it close to linear over most of the range but the variance is definitely not constant in either of these two plots.
Incidentally if you want to add a constant when taking logs, something a little above 0.4 generally works very well, but I usually just say 0.5; it's easier to remember and close enough.