An alternative to projecting the rasters at all would make sense if you plan to use them together and in different projections:
Add them to a mosaic dataset defined in the SRS the data is in. The footprints (and boundary) will then be defined excactly the same way as the original tiffs - so no need for NoData values. If you then use the MosaicDataset in the new projection (dont forget to set transformations needed in the MosaicDataset properties), there will be no NoData problem from the on-the-fly projection , thats made sure by the vectors of the footprints and boundary.
On the fly projection is computation, yes. On the other hand there will always be only one single resampling when using a MosaicDataset. Thazs much better than a resampled datatset due to the projection and another one due to the screen-resampling for display. Give it a try. For discrete data it will make a big difference, for orthos it would probably not be that important.