OK, Sorry to hear your problem.
You need to get that grout out, the new grout will only stick to the old grout. You can use a 'grout rake' or a machine similar to the '
Fein Multimaster' (using the Gold half moon head). But whatever you use, that grout needs to come out.
I'd say it's the tilers job to ensure the grout is fully out and make sure you check that it is and check for chipped tiles, if they chip any I'd also say it's there job to replace them at their cost and time.
Then select a decent grout, perhaps a flexible one as they have polymers in them to allow them to stick a little better as well as making it cope with any slight flex.
Hope this helps you and keep us posted on what goes on!