Hi Adrienne, 

yes this is definitely possible. See for instance the Biswal et al. 2010 PNAS paper for an analysis that combined the initial data released throught the 1000 Functional Connectomes Project.

However, beware, as can be seen in Figure 1 in that paper it is clear the center (and by virtue also imaging parameters) have a clear effect within the data. Despite this clear effect of center, you can still appreciate other effects of interest in the data (like for instance age, sex, etc.). We recommend modelling imaging center as a covariate in your analysis, but you could also do other things, like for instance investigate the effect of TR, volumes, etc.

So yes, its possible to combine datasets with different parameters, but be aware of them and adjust your model as you think might be needed.

To answer to your second question: I'm not aware that such database exists. The data included in the 1000FCP and INDI typically includes data from healthy control subjects, with no (available/released) assessment of their parents psychiatric status. Yet, as more data becomes available through INDI (especially more phenotypic data) I could see the data you're asking for being available in the future.

Best,
Maarten