Is there a place on the quad to put a backpack or bag?

Bags, backpacks or coolers can be placed in the basket or on the metal deck under the seats.

Back pack and small cooler can fit on the floor but only the size of 6 cans of soda or water small cooler

Yes

Yes. The carts have a basket for your purse or bag.

Yes. There is a small basket to place keys, cell phone, bottles of water :)

Yes. There is place to store a backpack or a bag. Something not too big is fine else it may not fit in the basket between the seats.