You don't say where you are, but I'm aware that some rescues DO rehome to households with small children. How old is your DD? Many rescues say no to rehoming where there are kids under 5, for instance. It's also worth remembering that rescues don't take this stance lightly and often do so because they recognise that combining very small children and dogs is not for the faint hearted and requires a great deal of commitment. It's not impossible and I know people who have successfully done it, but having slightly older children will make it much easier.
I work, but work from home. We did have a period where I was working full time outside the home - DH is a management consultant so works from home some of the time but also spends time with clients. On days when we were both out, paid dog walker came in. Rarely more than two or three times a week though. Some time after, we decided that it would be a better fit for our lifestyle if I stopped work full time (health issues as well) and now I work happily part time from home. We have also increased our dog numbers to three. 
Others have already indicated that they work part time, or do split shifts. It's also worth remembering that many families/households have a surprising variety of work patterns. These days, working from home is increasingly common - several of our neighbours do, one is in banking IT and the lady next door does something with telecoms from home. Add in various shifts, differing start times and it's possible to do it. For instance one friend works in a pharmacy 2.5 days a week (including Saturday) but her DH is a postman so he starts mega early and finishes early so dog is not left for very long. Teenagers also in the house.
Can't comment on cats 
Going back to your original question, it's doable if you get an adult dog, but simply not fair on a puppy.
If you wanted to PM me with your approx location, I'd be happy to look up dog rescues in your area and see if any did accept family applicants. I know DT and RSPCA generally don't but there are many other charities out there. Val will also have a very good picture of dog charities too.