You can remove the handrail in the middle when you put two staircases together. Just mouse-over the rail you want to remove and right-click, just like you remove walls or fences.

Your new floor plan is much better. I think you cut out just the right amount. Although I'd still watch if it were longer. I always loved renovating Maxis builds. Two suggestions for the kids bedrooms... 1) Quick & simple way to open up the floor space of the front kids room, which will mean less routing issues - looking at 31:56 in the back room, slide the chair & bookcase to the left into the corner by the door. Even swap them so the b'case is in the corner. Then simply move the front room's robe into the now empty space in the back room, directly next to & in line with the other robe but facing opposite direction into front room. Just like the robes for the master & cat room. Or 2) is a bit more involved as it moves the bathroom so it's between the 2 bedrooms which means losing the windows (common in American built homes) - looking at 32:37 move the desk out of the front room for now & place the front room's robe in it's place, turning so it's back to back with the back room's robe. Now move the bathroom back so it's against the back room's wall & essentially turn it lengthways so it touches the robes. The bathroom door would then be where the toybox currently is. And the front bedroom's door where the toilet currently is. Lastly, rearrange the bedroom furniture in the front room to your liking, remembering to place the desk back in the room. I hope that was clearer than mud lol. Looking forward to seeing this house in the let's play. :)