Neil - I should have also added that the root issue that you're seeing is that once you export the part out to an STL, it becomes a 'dumb' solid - i.e. all feature intelligence is lost. An STL has no mechanism for multi-body anything. The diagonal lines that you're seeing are a side effect of the export process. In terms of export formats in general, STL and SAT are about the most universal/neutral formats you can use - but that's because they force everything to be 'dumbed-down' to the lowest common denominator. IGES and STEP, in my experience, have a little bit more intelligence, and at least allow multi-body parts.
Sticking as much as you can with native AD files will give you the best chance of maintaining the feature history that you're after. Fusion does a pretty good job with Inventor files.
If you really want to preserve the assembly / part hierarchy, then jump out to the Fusion Web interface, and choose Upload>Assembly. This will allow you to pick all of the Inventor files, and identify one as the master assembly. Note if you're using Frame Generator, you'll want to dive down into the Inventor Frame folder, and pick all the individual frame members, in addition to the master and sub assemblies. Doing it this way will give you more usable subassemblies in Fusion.
AND..... the Holy Grail is...
If you use Fusion Team (kinda the Fusion workgroup/collaboration upgrade) then AD just came out with the Desktop Connector recently, and IT IS AWESOME. Still in Beta, and only available to Fusion Team folks for now, but it allows 2-way implementation of 'ANYCAD', which basically means I can roundtrip Inventor parts to Fusion and vice-versa. We'll design things in Fusion, and then insert those (live, not exported) Fusion files into Inventor, and use Frame Generator to fabricate the stand. Or generate parts in Inventor, and then our Fusion designers will insert those Inventor parts live into Fusion designs. Update the part in Inventor, refresh the link in Fusion.... everything updates. It works well enough - a couple of minor bugs yet to be worked out in Beta, but it's cool. Just FYI.
Todd
Product Design Collection (Inventor Pro, 3DSMax, HSMWorks)
Fusion 360 / Fusion Team