Yes this can be done.
When an Image Service (or Mosaic Dataset) is added to ArcMap it can be queried and searched. Users can make selection sets and by right clicking on the Image Service Layer and clicking on table open, they can view a table of the selection with associated attributes and refine the selection. IE user can easily select a range of raster datasets. If Downloading is enabled on the service then when users rightclick on the Service Layer, Download, Download selected rasters they can download the selected rasters. Either the complete raster or a clipped subset can be downloaded. This will optionally clip and download the individual rasters, without any resampling of the data.
Exporting is a bit different in that the server resamples (eg projects) and mosaicks together the imagery into a single raster. A user could export the imagery for different years by first locking the service to each specific year in tern and for each using Data, Export. The imagery can be locked to a specific year by using the lock raster feature. (Right Click on the service layer, Selection, Lock to Selected Rasters.) Alternatively create new layers for each of the selected rasters (Right Click on the service layer, Selection, Add Selected Rasters to Map). Each raster would need to be exported separately.
The same functionally can be implemented in web applications as all the required functionality is exposed through the REST and SOAP APIs.
