Like @amoeba, I don't understand your difference between Q1 and Q2: in any case, LDA obtains at most $k-1$ dimensions. For $k=2$ that's one dimension, for $k=8$ it would be 7. And of course, if the input space has lower dimensionality that that's the limiting factor.
One idea/heuristic behind LDA is that separating classes is easy if they are all spheres of the same size: all you then need is distance to the class means. Alternatively, you can say that the connection between class means are normal vectors for suitable separation planes.
In that narrative, you can look at LDA as a projection that in the first place does singular value decomposition (i.e. something very similar to PCA) of the pooled covariance matrix in order to get a space with such spherical classes. Let's call this our primary score space.
Depending on the customs of your field, this is either the PCA score space of the pooled covariance matrix = the PCA score space of the data matrix centered to the respective class means, or the PCA score space squeezed according to the eigenvalues/SVD diagonal matrix (some fields by default put this scaling for PCA into the loadings, others [mine] into the scores).
The primary score space still has the dimensionality corresponding to the rank of the pooled covariance matrix/data matrix centered to the respective class means. Pooled covariance matrix in this space is a unit sphere, i.e. each class is considered as a unit-sphere centered at the respective class mean, the means also projected into score space.
(At this point, we could derive other classification algorithms that use the SVD projection as heuristic but don't exploit/rely on the assumed unit-sized spherical shape of the classes.)
Now, as we have unit-spherical class shapes, the only thing remaining to care about are the class means. I.e., $k$ points. Even if our primary score space has higher dimensionality, the $k$ points will form a $k-1$ dimensional shape (simplex). Thus without loss of anything, we can further rotate our primary score space so that our $k-1$-simplex of class means lies in the first $k-1$ dimensions.
For the postulted spherical classes, all further dimensions cannot not help with the distinction: the classes have exactly the same size and position in these further dimensions. Thus, we throw them away.
(Again, you may derive another classifier that uses the first two projection steps but then keeps [some of] those further dimensions as the classes in practice may not be spherical.)
One classifer to look into when comparing LDA and PCA is SIMCA which can be seen as a one-class classifier analogous to LDA.
In SIMCA, you'll find the notion of in-model space and out-of-model space: in in-model space you detect changes of the same type of the usual variation within your class (but possibly of unusual magnitude). In out-of-model space you detect variation of a type that does not usually occur within any of the classes.
For our description of LDA, the $k-1$ dimensions we keep would be the in-model space, whereas the remaining dimensions are the out-of-model space. Interpretation will be slightly different compared to SIMCA, but you should have a start for your thoughts with this.
Best Answer
This is an excellent question because it touches on so many important concepts. The short answer is: Yes, this is possible, and can happen if your sample size is low.
Let us make the apparent contradiction a bit more precise. The MANOVA tests whether your data could have been observed if in reality there were no difference between the two groups (that is the null hypothesis). Your $p$-value $p=0.6$ is telling you that the answer is: yes, it easily could. At the same time, LDA results in a [almost] perfect separation between the two groups. So is it possible that in reality there is no difference between the groups but the actual data appears to be perfectly separable?
We can use a simple simulation to check. For various values of the sample size $N$ I generated random data from the standard normal distribution in the $80$-dimensional space $\mathbb R^{80}$ and assigned one half of the points to group $\#1$ and another half to group $\#2$. Both groups are therefore sampled from identical distributions, with true means at zero. For any value of $N$, MANOVA usually reports a high non-significant $p$-value, as expected. But let us look at LDA.
First of all, note that if $N<80$ then the groups can always be separated perfectly. Think e.g. of two points in 2D or of three points in 3D: however you assign them to two groups, one can always linearly separate them. On the other hand, if $N$ is huge, e.g. $N=1\:000\:000$, then it is intuitively clear that the two massive clouds of points (group $\#1$ and group $\#2$) will be entirely overlapping, resulting in no separation. But it might be surprising to see how slow the apparent separation is decreasing with increasing $N$:
The blue line shows mean values of classification accuracy for $N$ between $100$ and $1000$ (mean over $100$ repeated simulations), and the shading shows two standard deviations. At $N=100$ separation is almost perfect. At $N=200$ separation is around $80\%$. At $N=500$ it is still over $65\%$. One needs to get to $N>100\:000$ to get below $51\%$.
This effect is known as overfitting. Your $80$-dimensional space is large. LDA is looking for an axis with the best separability between groups, and when the sample size is not big enough, it can usually find some axis that by chance happens to yield good separability. That is why one should better use cross-validation to assess the performance of a classifier: if we used it here, then cross-validated classification accuracy would always be around $50\%$, as it should be.
Technically, overfitting happens because within-class covariance matrix cannot be reliably estimated with small $N$ (and so the sample covariance matrix in the example above will have some very small eigenvalues, instead of them all being equal). You might be interested in reading more in my answers here: