I think you are refering to all those little fragmented pieces that arent supposed to be in the trimmed facets. WHen you do the cross trim. you will want to make the trim curves "planar" and move them away from the surface a bit. This will allow them to trim the surface properly. AND what Michael said about making them be larger than the surface.

So if these were the trim curves:



I can go to the right view and use the edit frame to flatten (Planar) them out:



Then I would have a configuration like this for trimming:



WHen I run trim on these planar and extended curves. they trim without those little fragments:




Sometimes you have to work your trim curves a bit because MoI does this kind of "automatic projection/extrusion" thing and this can cause these self intersecting anomolies when the trim curves kindof float in 3d and trim in "Multiple directions"...

Here's your file back just before the trim. I just flattened out those trim curves.