Boringly I can answer all you’re questions.
I take it the floor is a click system. You’ll need some wood glue, ideally MDF specific. A square tapping block, if you can find a bespoke one a square wood block half the height of a house brick will do. A stanley knife. Ideally a table saw but if not a jigsaw with laminate blades.
Usually start on the longest wall. Remove the rear tounge from floor where it backs against the wall and allow 6mm minimum expansion gap and add spacers or weges to maintain the gap. With regard to the breakfast bar you’ll have to leave a gap that’ll need covering. Too risky to go tight in a high moisture content area like kitchen. If you want something contemporary looking maybe use square section MDF, prime it and spray it gloss white.
Going under the plinths with the floor will require you to remove the ridge from the tounge side/end (Stanley) of the floor and gently tap it together (tapping block). Use glue on the removed tounge. You’ll have to do this as the click system of the floor requires a 15 degree tilt to engage.
Anyway enough of that. Good luck!
Edit: take your time! Can’t stress that enough. It’s a big puzzle and the time taken to think what is going to happen two rows of floor over is key.