I would:

- first mosaic them,

- then project them to the chosen projection

- and finally cut the extent to the chosen extent.

I agree with Klemen and would add that in dependence of the GSD (pixel size) of the data you want to correct, you might think about resampling ASTER GDEM data to have matching GSDs. In this case you would avoid artifacts.

I actually have an IDL code that does that what I described above and projects everything to the UTM projection over chosen extent with chosen GSD as Ulf said.

Thanks for the tips! it is done

what about the tool Set Null and Cut an fill, are it necessary process or not??