Given your space constraints, I think it would be best to hang the TV on the wall that is common to the study/living room (just to the right of the door entry). You could get two or three small upholstered chairs for the living area (gives you flexibility in moving them around.) I would have a table surface hung so it can swing out and be used on the wall between the door and the opening to the kitchen. If hung with a hinge attachment so it can be flat on the wall when not in use, you still have space to move around. Backless bar stools could be lined up on that wall when the table is stored flat to the wall, and still be used for more guest seating.

Let your wooden beam be as it is. put your Led lights in the other direction ( two on each side in one rectangle , so the total will be 16 units of say 10 watt each Led.), as to balance the horizontal and vertical shape. Don't put anything on the wooden beam to retain its the beauty.

What are those red curved lines? The measurement is also not clear. What i understood with this is that , You should have two different rooms - one is formal drawing room with only sitting area and for that you need not provide a partition as it should look grand from your entryway and second one should be the living room. There you must give a partition between both the rooms so that the person sitting in the drawing room cannot peek in the living room and it'll be kind of privacy as other area will also be not directly visible . By giving a partition you are messing up the passage way more.

Hello, the city is Gurgaon. I have not yet decided on budget exactly yet. I am exploring my options and will limit it soon. Consider it to be a low-cost approach, as a starter. I have attached the picture of measurements. Can you please elaborate what you mean by "layout"? Sorry, for this being a noob question!