What happens to the baseboard heating vent when you remove the wall?

Good question. The contractor said it could be moved or removed. I am also worried about how the floor will be matched between the rooms

Not understanding the "half wall" comment -- hard to give advice without knowing what the rest of the layout looks like. Right now the wall and doorways control the flow of traffic in a certain way. If/when you remove the wall, what will traffic patterns be like? Let's assume you can successfully relocate the heating and solve the flooring issue...appears that you have pretty large rooms, and will have the opportunity to arrange and rearrange until you find a furniture layout that works for the space. You might consider finding an on-line app or use a furniture store's app that allows you to put in a room's dimension and then put in furniture and move it around -- it's kind of fun.

Sorry, when I mentioned half wall, I was actually referring to the vertical wall that would remain in the living room to the right of the beam. The flow of the room would be hallway to living room and into dining room and the leading to kitchen. The contractor mentioned about removing the wall between living room and dining room by cutting it rectangularly across, not all the way to ceiling. This would leave a left angled triangle on the ceiling. Does that seem like a good idea?

Before you do anything please get another opinion about that wall you want to remove being load-bearing. It looks to me like it could easily be supporting the beam that runs through the living and dining rooms. The best way to deal with the flooring is to weave in new wood and then refinish the floor in both spaces.

Clarification on opening. The wall would be cut straight at the horizontal line to the left of living room painting and to about the edge of the painting. Meaning, the right side of the wall past the painting would still remain.