Messonghi has a very safe beach, gently sloping and ideal for children.
Yes Dimensions is most definatly still there and continues to be run by Spiro and Anita. They were featured in the Daily Mirror's item "The Real Shirley Valentines" last July.
The Blue Lagoon trips run from the south of the island, definatly from Messonghi but I've also heard they go from Kavos too. They don't go from Sidari I'm afraid as that's in the north. They'll be plenty of other trips running from there you'll find.
As to the cost, can't be certain this year, but it won't be high if booked direct rather than through a tour operator.