Once you've paid to go in the gardens, can you leave to get food and then go back in to picnic??

Should be fine because they give you a sticker to show you paid for entry

Not sure but you can ask in the Visitor Center

To my knowledge, you can get something from the counter for reentry. I have also seen people eating picnics in the park.

I don't know for certain, but thru seem pretty laid back. I'm sure if you asked them it would be okay.