I've always had them. I had a JRT when both girls were born. The dc were 3 and 6 when she passed away and we ended up with our current scruffy terrier.
It's not always easy fitting in dog walking and training around young children, particularly when they are ill or it's raining or CBeebies is still on air or there are pictures that need colouring in etc. and the last thing they want to do is join you on a hike around the local nature center.
Teaching the children how to behave around the dog is far harder than teaching the dog how to behave around the children.
Any toy that is left on the floor and subsequently destroyed by the dog will immediately become "my favourite toy" even ones that are rarely played with and there will many tears, loud howling and you can be certain that they will let you know it's your fault.
If this is the first dog you've owned I would strongly advise you to wait until your youngest is in school. Please believe me when I tell you you are underestimating the amount of time and money your dog will demand.
I did and I was brought up like my children. I've never not had a dog, from being born.
If you can't wait make sure you have plans in place for people to walk the dog or look after your youngest while you walk the dog if a) the weather is absolutely horrendous or b) the child is sick with more than the sniffles.
Make you sure you keep an empty credit card for emergency vet fees (you can claim this back on insurance but not all insurers pay the vet upfront and not all vets are happy to wait for payment from the insurers so it's safest to have access to funds immediately if you need to).
I know I keep posting negative things on your threads, it's just it's coming accross as a very impulsive descion on your part and not something you have fully thought through or planned for. As I said before owning a dog is about far more than just wanting one. I want a little monkey. I'd love a little monkey. But I can't have one because it would not fit in with my life style and it would be impractical and I'm fairly sure once I got a monkey I would fully understand just what a silly idea it had been.
And please consider what you will do if your LL serves you notice to quit. How/where will you find another pet friendly rental? What will happen to the dog if you can't find another pet friendly rental?