I must ask...why the highly unusual sprinkler cycle?

That wouldn't be enough to dampen the mulch in my garden areas, let alone soak into the soil/root system.

It sounds like you mean that there's a sprinkler already installed that will go off twice a day for 5 minutes. I don't think this will cause problems for your bed. Give your plants adequate spacing and you can grow pretty much anything that's suitable for your area. As rhizo_1 says, that amount of water won't amount to much, so during hot weather you may need to water the plants yourself.

Make sure the sprinkler isn't mixing in fertilizers or pesticides.

What do you mean by small plants. Rosemary can become quite large and oregano can spread for as far as it will be allowed to reach. Chives and garlic chives like to be planted in the fall. as do most herbs. What ever you plant should be fine this far south except those that do not like our summers.