Hi Danny, nice to see you, come, join the ‘fun side’ of the beach 
Hmm, just get a black screen, nothing, no shockwave logo, no asking to update shockwave (win 7, firefox), works ok in IE.
Flash animations: Worse case I don’t see why you couldn’t do the heart as a mesh, maybe using a fancy shader for the flat shading and outlines. Perhaps some of the 3rd party vector solution can provide this level of vector graphics, i’m not sure.
Animated 3D meshes with flat colours: Should be very simple. Unity supports actual 3d file formats such as fbx, obj etc as well as Max or Maya as long as the 3d modelling software is also installed on the same machine, though its usually recommended to export to fbx. Animation is supported as is, no messing around like in SW3D.
3D with overlaid Flash graphics: Difficult to say but again 3rd party vector support should be able to do most of what I saw here - nice refraction demo btw.
2D sprite animations: Didn’t see anything here that simply switching to 3D (spheres) wouldn’t be able to accomplish. I’d also argue that you’d find Unity physics far easier and more usable than Directors in this case. However when you really need 2d sprites there are 2 or 3 really good (judging by the comments) 3rd party solutions for this.
2D graphics animated on the fly: Couldn’t really identify what or why imaging lingo was needed here? For the most part I would say this could be accomplished using just 3D and Unity particle systems (which are very advanced and even allow you to over-ride particle positions, so you get the benefit of performance with exact control).
If you do need imaging lingo type functions, then Unity does allow you to update textures. Its slow due to updating a texture, but perfectly usable as long as you aren’t updating lots or massive textures each frame. No fancy ink effects though so you’d probably end up writing your own imaging lingo replacement, but it should be pretty simple. Just make sure you always use power-of-two textures and you ‘should’ be ok.
I suspect this is going to be one of the hardest parts to determine. Pretty sure Unity supports multi-byte, right to left text, but have no experience of it and judging by the comments here, neither do many others. This might be a question better aimed to Unity sales.
As for the rest, vectorised graphics are problematic, in that they are not natively supported, but there are some rather good 3rd party solutions for this. Just be careful that they are not native code plugin solutions as they wouldn’t work in the webplayer.
e.g. RageTools looks interesting
https://www.youtube.com/watch?feature=player_embedded&v=PlYvJ9BU3AA#!
Honestly as others have said your best bet is to choose one or two of the more complex featured demo’s you have, get Unity Pro free for 30 days and try to convert them over. The community here is generally very helpful, though a bit 'game centric (in my opinion) as is Unity itself. I’d also suggest going for C#, its what Lucas recommended to me and apart from an initial learning hump, I really do love it.
If you really get stuck, simply don’t know where to start, or perhaps want a crash course around the Unity IDE, feel free to send me an email. Perhaps we can set up a Skype share screen and I can talk you through some stuff to get you up and running quicker. Unity really isn’t hard, but it is ‘different’ to Director. For example Unity asset bundles are a poor substitute for Director castlibs.
Conclusion
However as others have said, if your projects are predominantly animated vector graphics, then you may just be better of going with Flash and hoping Adobe doesn’t decide to abandon it like they did with Director. I don’t think any other solution will come close to providing the level of support for the media files you need. Unless you can embrace 3D doing this in Unity may just be more trouble than its worth.
I might even question why bother switching from Director at all, if only because you do seem to have a huge amount of content to convert and frankly no IDE is going to be a perfect solution or have the risk they might disappear one day.
I don’t envy your position. I was lucky the majority of my work was 3D by the time I ditched Director so it was an easy choice.
