Is there a technique for flattening out the carpets that seem to have such a bad habit of curling up/wrinkling?
My 1979 280SE blue carpets are still good looking accept for this issue.

I was wondering if steam-cleaning then 'pressing' them somehow (laid on the floor with a piece of 3/4" plywood laid over then stacking anvil, vice, other heavy objects on top) would have a positive effect?

Or pealing them off the insulation then applying some sort of semi-rigid backing? (gluing them to light gauge corruplast?)

Or am I just dreaming up stupid ways to waste my time?

Also, what is the preferred method of cleaning carpets?

I would greatly appreciate learning from a professional or well experienced DIY'er

Are you dealing with the carpets fitted to the floor pans or the floor mats? I fixed curly floor mats in my SDL by cleaning them in the bathtub with a mix of detergent, rinsing well, then laying out in the sun on the driveway. They came out all nice and flat, clean, and smelling nice!

For the carpeting inside the car, you'll have to get it hot and press it into place. You might try a clothes steamer and pressing in place until it cools. You need to relax the fibers and hold them in the position you want them to assume.

__________________
Current stable:
1995 E320 149K (Nancy)
1983 500SL 120K (SLoL)

Black Sheep:
1985 524TD 167K (TotalDumpster™)

Gone but not forgotten:
1986 300SDL (RIP)
1991 350SD
1991 560SEL
1990 560SEL
1986 500SEL Euro (Rusted to nothing at 47K!)

You're method is encouraging and reassuring.

My inclination is very similar in that the carpet backing fibers need to be destressed