Gorgeous space!!!!
Where are you thinking to add a rug - under the sofa definitely

By the table area - not necessary
Runner by the stove can be nice but not necessary
Runner down the hallway also can be cozy

Wondering if you taped together Kraft or Butcher paper, which might help to get a sense of where a rug(s) might work.

Yes, I think the space would look great with some extra texture in it. Where to add them, I'm not sure. Playing around with butcher paper is a great idea.

I love your space! It’s so chic. Maybe under the sofa, but we need a pic of that area.

I love your space! I feel your space has a neutral and relaxing vibe. I love color so one option, depending on what colors you like, would be to add a runner long and wide enough to be centered in front of the island coming into the living room. Can't see your couch and whether or not you have a rug in there, but if you do, it should coordinate with the runner. For me, a neutral and easy choice could be jute. A warm beige jute rug would add more warmth. Plus niw a days there are more design options for textures and prints on jute. A plain jute runner and a printed jute for the living room would continue the neutral color tones and add more warmth. Also, I am a plant person and a few liw maintenance plants like a tall sansevieria by the console or sone table top pothos on the console could help with adding color. Just make sure if jute is a possibility make sure it's quality. Another option could be an abstract print with creams and maybe some colors resembling your island.