Dear gentlemen, thank you for interest, comments and valuable suggestions about this fungus. I start tackling all your questions here, and micro in the next post.
After a hard day at work, and sipping down my coffee, i was flipping Zannichelli fungi d'italia and spotted a fungus similar to mine - Guepinia helvelloides (sp. #1545) which is the same as what have been kindly suggested - Tremiscus helvelloides, but then the size (3-12cm) the habitat (mountain) and substrate (wood debris) did not match at all. This is about 1-2cm
Quick answers:
Is it growing on wood in the ground? - I did not see obvious wood parts in the compact calcareous soil, but it was growing under bushes of Pistaccia lentiscus (woody perennial) and it cannot be excluded to have small decaying twigs in the soil - honestly I don't think it is the case.
Dry at 35C for sequencing: Yes I can
Revisit site: It is in a remote place but I can
Infected: No it is healthy
Mature specimens: Those at home no 
Is T. helvelloides specific to wood : On the net I see some images of this species surrounded by moss
What does it match : The salmon-apricot colour, the fine pruinosity, the attenuated stipe coloured pale yellow or white, the wrinkled unsmoouth surface.
Microscopy : yes, see post below