Ok lets take these one at a time.
1. "Transfer VHS tapes to DVD - straight transfer no editing."
You will still need to capture and encode the video. If your camera has an a/v input that will allow you to run your vcr into it and out to the firewire that would be great. You could use adobe to capture it then any dvd software you prefer to encode it and burn it. I use TMPGEnc Pro to encode. Then Nero to burn. You could use the new adobe dvd software. I have not seen it or used it yet, I like to try before I buy and have not found a demo or friend that has it yet.
2. "Capture the output of Sony handycam playback to allow for video editing - lip sync is a must. Output to mpeg and DVD."
Same here. Does your sony have firewire? If it does just use adobe to capture. No worries, I have not had any lip sync problems. The lip sync problens usally occure when the video is encoded. Make sure you use a good encoder and run it and it only. Don't surf the net or anything while its running. Just let it do its job. Your camera may offer usb capture (mine has both).
3. "Creation of video content from computer (graphics, slides, screen captures etc), add voice overs, splice in other video and output to mpeg and DVD."
Adobe will allow you to use alot of different graphics formates.
I have used it to make a video of pics with transitions, music, backgrouds and the like. It turned out real nice. I have also used it to produce a weekly TV programe for a church that was on a a local station. I used alot of motion backs (Digital Juice, etc.) and grapics, titles, and so on. A screen capture is just a bitmap or jpeg. Just save it as a file you can use, like jpeg, tiff, so on.
4. "Is .avi the best format to manipulate video + audio and stay in sync?"
When you capture Digital Video it is in DV formate which uses the extention avi. Yes, it is best.
I would go for the 8890, I use my 8887. The tv tuner is not the best way to capture the video. us your camera to conert it to firewire. Adobe has a new version of premeire out called Adobe Premeire Pro. I have not used it yet, but it looks great. Here is a web site that may help you.http://www.dv.com/
You will need to register to get in to all the content, but it is free to get in.
As far as an externial box there are alot to choose from, but if you go with the 8890 you want need one. one less thing to hook up.
Hope this helps. Don't get frustrated. Keep working at it, you will be editing like a pro in no time.