Your choice of mosaic dataset is a very acceptable way of handling lots of image files, I use them frequently for just this sort of thing and have found them to be as useful as a single, mosaiced, raster but much quicker to create. You can also use rasters located in many different folders to create a mosaic dataset without needing to copy them all to a single location, a real drive space benefit, as only the overviews need to be stored on the same drive as the mosaic dataset.
The most serious drawback to using this raster storage method is that only Esri software can access mosaic datasets. Should you need to create a mosaic-dataset-like object which is usable outside of Esri there is the GDAL VRT dataset which is an XML file recognized by Esri, QGIS etc.. One of these can be created with GDALBuildVRT. This virtual storage type is significantly slower at redrawing and exporting than a mosaic dataset so I would only recommend using it outside of Esri; it is possible though to link the rasters in both a mosaic dataset and a virtual dataset (VRT) as neither format is exclusive.
You can handle exporting your mosaic dataset to another format with a reduced extent in one of two ways: geoprocessing environment Mask and/or geoprocessing environment Extent.
If you have a polygon shapefile you're interested in you can use both the extent and mask to reduce the size of the export and to trim off any areas you're not interested in within the extent rectangle of the feature class. Otherwise, if you're only interested in exporting the raster that is overlapping existing data or a known extent you only need to set the Extent environment, either by specifying an existing feature class or raster or by manually entering the extent to export within.
I have found it best to ensure that all coordinates for extent and mask are in the same coordinate system as the 'to' raster.
If you are overlaying your output raster with another look also at Snap Raster and Cellsize to ensure pixel-by-pixel alignment.