My old house had 2 planted real close to each other in the front yard. When they get big, very little sunlight hits the grass so it starts to thin out. You also have the problem of the branches growing into each other, so you end up trimming them a lot. So yes, IMHO, they can be planted to close to each other.
Don't know what your turf is, but Bermuda will not grow under the live oaks. Palisades Zoysia, a beautiful turf, will grow there however. You may need to switch your turf type.
When you have multiple Oaks, you have to water your lawn accordingly, meaning those Oaks are soaking up some water that your grass would otherwise have available. So you must water longer in those zones. Trees get thirsty in the summer and can hog the water.