Ken Bollen and I wrote about negative variance estimates (aka Heywood cases). You might want to take a look for some insights. For this huge model, God only knows how model misspecifications are going to show up, but in my experience, Heywood cases are typical outlets for the model to let the steam out when something is not fitting right.
That having said, I would try different diagnostics: first fit all of the submodels with 6 or so indicators, and see if there's anything wrong with them. In the CFA context, I would imagine that underidentification would arise only if some variables have zero coefficient/covariance with the factors they are supposed to measure. You should be able to catch that with the analysis of subscales.
Finally, for the Likert scales with 4 categories, you really should use polychoric correlations (polycor package). For one thing, the categorical nature of the data would yield the likelihood ratio tests unreliable (as if I would trust that 900 observations could give rise to 2166 independent degrees of freedom, anyway).
I think your conceptual understandings of reliability (via Cronbach's $\alpha$) and convergent validity are correct. However, I believe that the way you have defined evidence for convergent validity is mistaken. Your reflective model implies that these six items are manifestations (i.e., caused by) of your latent construct; to then use these same indicators as "...other measures that it [your latent variable that is presumably causing these indicators] is theoretically predicted to correlate with" seems very circular. How can the variables be considered manifestations of your latent variable, and "other measures" at the same time? Instead, I think you should be establishing convergent validity via inter-construct correlations, much like you would discriminant validity.
Two other quick thoughts:
1) I've not often seen Cronbach's $\alpha$ calculated for latent variables. Rather, Cronbach's $\alpha$ is often calculated for scale scores (averages, sums) that are observed. You might be interested in calculating construct (or sometimes called "composite") reliability (Hatcher, 1994), which can be done with the following formula:
($\Sigma$$\lambda$)$^2$/(($\Sigma$$\lambda$)$^2$+$\Sigma$$\sigma$$^2$)
where $\lambda$ is a standardized loading, and $\sigma$$^2$ is a uniqueness.
2) Your AVE seems similar, in concept, to the calculations for how much variance (similar to the previous formula) is explained by a given latent variable. This calculation could be taken as some preliminary evidence of construct validity, as if your latent variable is not explaining a substantial amount of variance in it's indicators (e.g., >.5), then perhaps it is a poorly conceived latent variable:
($\Sigma$$\lambda$$^2$)/(($\Sigma$$\lambda$$^2$)+$\Sigma$$\sigma$$^2$)
Best Answer
There's nothing special or magically different about structural equation modeling (SEM) and other statistical techniques. Regression (and hence t-tests, anova), manova, etc can all be thought of as special cases of structural equation models. In addition, SEM and multilevel models are often equivalent - see http://curran.web.unc.edu/files/2015/03/Curran2003.pdf [edit: link updated, thanks @Peter Humburg) . If something is an assumption in statistical analysis generally, it's an assumption in SEM. If something is an issue or a problem in statistical analysis generally, it's an issue or a problem in SEM.