WRT the oil temp for takeoff
1) Rotax specify that as a minimum
2) I have heard that this is because there is a bypass valve in the oil filter that remains open until the oil is at 50 degrees
3) I masked off a third of my oil cooler because my oil temps were too low during flight, but especially on descent. Now, my oil temps run in the middle of the green arc during flight, climb to the top of the green/bottom of the yellow on a sustained climb and sit at the bottom of the green on descent. Clearly, the Rotax supplied oil cooler on my airplane is providing more cooling than my engine needs. I have a simple normally aspirated 100HP 912ULS.
As for not doing runups before takeoff? Not this little black duck. I do them every time. I want to know that both mags are sparking before I go hurtling down the runway with a fence at the end of it.
WRT to cooking the engine? IMO, if you're keeping an eye on the temp gauges, you shouldn't be cooking the engine. I couldn't vouch for air cooled engines, but the Rotax doesn't cook like that. I have had an occasion with a warm start on a very hot day when my coolant temp rose above normal, but as soon as I started my takeoff roll, the temp came down about halfway down the runway.
So, how a pilot treats his own aircraft with Lycomings is his business. That it "doesn't use a drop of oil" says only that his engine is still in reasonably good condition, but it doesn't logically follow that his lack of runups is the reason. That's a bit like saying that because rain comes from clouds, if there are clouds in the sky it must be raining.
Finally, WRT to the long warmup time for the Vixen, yes, he probably had a similar setup as most Rotax engines. I think the standard oil cooler has been designed to cope with the higher power and turbo charged engines. With the standard oil cooler on a normally aspirated 80 or 100HP engine it can take a while to get the oil temp up, especially in the winter, but it's still required by Rotax.