You've essentially asked and answered your own question. "Is the shut-down and start procedure too lengthy?" Yes, sometimes. "Start cycles on engines?" Definitely, particularly on short sectors. You could also add; Lack of ground-power for re-start, this is important at altitude. Lack of technical support in case of difficulties. etc
"Does it make a difference whether it is a turbine or reciprocal engine?" I've not worked with recip-powered helicopters so can't give a definitive comment, but I would imagine it's pretty much the same. I've seen turbine-powered helicopters sit on the ground for 20 - 30 minutes with their engines running, all in a days work on some operations.