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Dear Sarb,
1. The F-contrast (effects of interest) shows only few small significant
regions at p<0.05 (uncorrected). However, if I look at the contrast c1-c2
(i.e. 1 0 -1 0 0 0), I get highly significant effects in those areas I
expected to see. How is it possible that I don't find these regions in the
F-map? (I think it is important to add that the contrast 0 0 -1 0 0 0 did
not show a significant effect either).
I recently encountered similar problems in one of my analyses (sparse and insignificant effects when I looked at the F spanning the space of all covariates, yet
significant effects when I examined a specific combination of those covariates). I think this can be explained by two factors: the difference between a t  and a F contrast on the same data, and the loss of degrees of freedom when you use an F contrast with n different covariates. In a comparison between a t and an F, you need to double the p value of your F to ensure a similar level of significance: so if your t contrast has a p of <0.001, your F should be <0.002 ( this is because a t is a one-tailed test, while the F is two-tailed). The loss of degrees of freedom results from your F examining the variance explained by an n-dimensional space spanned by your covariates. The t looks at the variance explained by the one-dimension spanned by your contrast: so, due to this constraint on your anlaysis, the significance of your effects increases. If you want to ensure that your F and t are comparable, construct an F which has similar degrees of freedom to your t, and adjust your significance levels at which you display it accordingly (ta to  Jesper A. for tidying up the preceeding explanation!).
2. If I want to plot the adjusted data of the areas showing significant
c1-c2 difference, it appears that this is not possible. I get the message
"no raw data has been saved at this location". I assume that this has to do
with the fact that there is no significant F-contrast at the same location,
and therefore no raw data has been saved there. Is there a way to plot these
data?
As far as I'm aware, the timeseries data for these voxels is not retained, so all you can look at is the parameter estimates. I think you need to repeat your estimation, but change the F contrast threshold in the spm_defaults menu (it's in the 'defaults' button, under 'spm statistics'.) If you make this more lenient, your voxels should survive the cruel scythe of the F cut-off.

Best

Dave