Hi David,
(hi and low answered most of your questions while I was typing.. so some duplcation below)
I own and use both an R44 and DR680 (and DR2d). Both have worked very well for me. I'll try and answer your questions and add some of my own insights.
1) Can an single input be mapped to multiple tracks with different gain? Specifically, this is to let a track be recorded comfortably below clipping to fix problems in post. The Marantz 661 has this (Mono-D, 20db lower) and I think the Tascam DR-2 (dual recording, 12db lower). If the answer is no, can the mix track be used?
No. DR2d can do this (not limited to only -12dB, you can choose from several level differences). You can route a copy of two tracks to the mix track via the monitor section, panned hard Left and Right, but that doesn't really do what you want as an input level safety because if the primary channel clips the mix will too, and if input gain on the primary channel is set deliberately low any gain addition made for the mix channel is done after analog-digital conversion in the digital realm.
2) Are the outputs live without recording? Reading the manual, it looks like the Pause bottom arms recording and then the tracks have to be selected. Does that send their output out?
Only the digital input (which I just discovered last night). All analog inputs require the recorder to be in rec/pause and the track in question armed to pass signal through to the output. The output is selectable, with a choice of either individual channels or the stereo monitor mix.
3) Is there a way to connect a lav or other mic with a 3.5mm jack that requires 5v power? If not, I think I have a battery powered adapter, however that is additional cabling complexity and another thing to go wrong.
For 5V mics like lavs I prefer to use phantom power adapters from Niant (owner is a member here). His PFA adapters are built into XLRs that plug directly into the recorder inputs, come in sets of two, work great without extra battery worries and are inexpensive. I use these with both the R44 and DR680 and recommend them highly. They can have either a 3.5mm input jack(s) or mini-XLR(s) on the mic side.
4) Can the digital output be fed to a computer? I'm thinking the mix to the digital out to a USB box that takes S/PDIF input. Is there any other way to connect it?
Sure, that works, the other 'real time' transfer method would be using the analog outputs like older analog gear. Most users simply transfer the files to the computer by using the USB connection for file transfer or taking the card out and putting it into a card reader.
5) Are the switches on top easily accidently activated?
More easily than I would like, (and like aj mentions above, more easily than on the R44)
6) Is the Tascam case best or is there a better option? Does the Tascam case work ok with the cables? Does it have a battery access door?
Access to all ports and doors, it's small and protects the machine only, no problem with cables. I use it in combination with a larger bag that can hold the recorder and the necessary extra gear.
7) Does this recorder have a lot of plastic for this price level? Does it hold up?
Typical construction for it's price level. Actually it has no competitors at this price with the same features. R44 is closest and similar.
8 ) Initial owners reports several hardware problems. Have those problems been eradicated?
I've had no issues as have many others, including using the digital input (output from the R44). Some have had problems and expressed concerns about Tascam's service department and handling of warrantee issues. It seems like Tascam payed attention and has stepped up to take care of most(?) of those after the recent problems thread bloomed here, but I haven't really kept track. Other's such as aj have had less than rosey experiences and are understandibly soured somewhat.
I like both the R-44 and the DR680. If all you need is 4 tracks total without more advanced features like the ability to pan each channel output to the monitor section or the ability to 'link channel gains' across sets of inputs to keep gains identical and adjustable as a group, the R44 is slighlty smaller, simpler to use, generally less complex and very well tested and liked by users. If you need more than 4 channels or some of those other features, the DR-680 offers that.