You've mentioned that the house is new. Have you approached the builder with these 2 items yet? What do they say? Do you have a warranty?
The grout may be cracking for a number of reasons, none of which are limited to the grout setting too fast or not properly mixed or an unstable subfloor. If possible, tell us what you have for a sub-floor and what the builder has said. From there, perhaps someone can be of further assistance.