DEMAT account in ICICI is safe?

You are absolutely right, ICICI is just a kind of a go between, your money is with SBIMF and Sundaram. I do not know about the safety of MFs because there is always a risk involved in such things, but under the present conditions, I would say that both are safe, and you don't have to worry about ICICI going bankrupt or anything, the second largest bank in the country is not going to be bankrupt that soon, quit worrying, just concentrate on how your MFs are doing.