I would like to shoot the Andromeda galaxy with my 70-200 F2.8.
I do have a star tracker (not soo precise).
In general for faint things it is better to use a wide aperture, to gather as much light as possible. but given that the galaxy is RELATIVELY small with a 200 mm and I have to crop a lot, would it be better to shoot it at the lense's sweet spot to get maximum sharpness, and take more shots to combine in deep sky stacker? or will i miss all the very faint details?
There are at least two ways of looking at a lens' sweet spot. One is in terms of absolute acutance when measured after very careful manual focusing on a well lit test chart. Another is in terms of Depth of Field sufficient to cover errors introduced by autofocus inaccuracy. But DoF will affect results due to focusing errors much more at very short focus distances than at very long focus distances such as several million light years. At such long distances the accuracy of the focusing matters more.
In the case of astrophotography you're almost certainly using careful manual focus because AF doesn't usually work with stars and deep space objects such as distant galaxies. With the short throw of the focus rings on modern AF lenses the variation there between your best and worst manual focusing efforts will likely far exceed the difference between the lens' acutance at f/2.8 and at f/3.5 or f/4 or f/5.6 or wherever the absolute sweet spot of the lens is in term of aperture.
So until you can consistently focus your lens at deep sky objects with a very high degree of consistency what aperture you use really doesn't make much of a difference in terms of absolute sharpness. Unless you have a very precise tracking mount the shorter exposure time afforded by the wider aperture will likely lead to better results than using a longer exposure time with a narrower aperture and allowing tracking errors more influence on your result.
That really depends on the quality of your lense.
Since you have a tracker, and assuming you don't own the very high end of lenses, I would suggest closing the apperture to just 3.5. If you are able to do exposures of some 30sec, that might give you good results! (At least for me it does, with a comparable setup)
