you should try round-up it is the best ( and most expensive) weed killer . it is child and pet safe but i recomend keeping them of the area for an hour or so .
yes, Round up or its equivalent fits the bill. Use it once the plants are actively growing and make sure it is not about to rain just before you do it.
One I can recomend, which has the brand name "GLYPHOSATE" Comes in a handy ready to use, 1Ltr bottle with a spray trigger and can be used between march-sept, which is a systemic weedkiller specially tailored for paths and patio's and is safe for pets and children once dried. One application a year is usually enough..
Try using household salt with hot water .. better for your dog and the environment. Either dissolve loads of salt (buy the cheapest) in the hot water in a jug, or put salt in between the crevices of the patio, and then hot water.