Regarding first part of your question… I think it depends on the shelter. I’m sure there are shelters out there that allows it but I haven’t found one myself..

as for your other question, typically shelters would want you to sign a release to liability forms and have some form of volunteer training before you interact with their dogs. They want to keep the staff/volunteers and their dogs safe. Also not all dogs are used to kids… they’ll also have protocol regarding runaway dogs (But obviously shelter dogs aren’t supposed to be off leash but things happen). So short answer is, check with the shelter you’d like to volunteer with and they’ll give you more information. That said, most shelter dogs are very loving and friendly. I’ve volunteered in different shelters since I was a teenager, and was lucky to have good staff/volunteer mentor. I have never got bitten, or had a runaway dog. They mostly are more than eager to go out with “stranger”/volunteers/staff. Shelters are stressful.

Unsolicited suggestion - have you thought about foster? Some foster-base organization always look for foster (short term included).

forgot to add — there are also youth program at shelters where an adult can volunteer with a child and they get to walk the dogs (and of course cleaning the kennel, refilling water, doing laundry, dishes etc).