Can the various diesel aviation engines (Thielert, etc.) burn this sort of diesel safely, or are they limited to Jet-A?
Nope.

The Thielert 1.7's are certified to burn diesel in Europe, but the diesel used in North America doesn't cut it, they're only certified for Jet-A here.
IIRC, this is due to the tendency for regular diesel to gel when it gets cold (ie higher altitudes). I've heard it's relatively safe to put regular diesel in the main tanks on something like a DA42 because enough of the fuel is being circulated through the engine to keep it warm provided it's not super-cold out, whereas the aux tanks would need Jet-A. However, it's a tad risky anyway and certainly not approved.
On the off-road diesel, it's my understanding it's not filtered nearly as well as #2 diesel. That could cause problems with filters in whatever vehicle burns it, particularly if it's higher flow. The diesel Jetta I drove, if the filter got dirty enough it would suffer horribly from fuel starvation.
Actually, it's exactly the same, except for the red dye. John's got it right:
Yes, the red dye is off road use only because the tax is not added to the price. The red dye is used to catch cheaters.
As for your tax money, if you were to use fuel taxed for highway use in your airplane, I think you can get your money back by using IRS form 4136 or form 8849 schedule 1. I'm not an accountant so don't ask me what the heck the difference is between the two!

I'm sure there's another form you have to turn in to report your aviation use of the fuel and pay your share of the FAA's budget as well.
FWIW, not too many places sell red diesel, but I saw it in Grand Island, NE the other day at $2.759/gal.