Would it be better to put up plexi or glass above the railing and use this for the play room, so he could have a space where the door could be closed?

The first contractor you need to meet with may well be a heat/ac guy -- where are your supplies and returns for the room. Do you need to install an in-wall heat/ac unit such as are often found in motel rooms or hospital rooms or assisted living rooms where individual control of the room temperature is desirable.

If, as it appears, you will be closing off the skylight that provides natural light as you close off the room, is a window that opens a preference or do you need a window for … possible escape route because of a local fire code? Would adding another skylight be more practical than a standard window?

One thing to check on is whether or you could install a GLASS wall that will mute or at least muffle sound but let the natural light from the skylight shine into the space.

If you're rethinking lighting, something to consider might be to add a ceiling fan light -- especially if the room tends to be difficult to cool. A fan something that could also provide a bit of white noise.

There are soundproofing panels that you can buy. Expensive, however, if someone is trying to make a living from home and in need of quiet to do that, the cost of the remodel to close this room, including soundproofing panels, should be tax deductible -- just be sure to keep ALL relevant records, including for all expenses.

Does the staircase lead to more than the one room you want to close? If not, closing off the staircase and adding a door both at the top and bottom can help muffle sound.

Do keep us updated on your progress.

There must be glass that mutes sound -- think … recording studio. If it is within your budget, use something clear such as that above the half wall near the skylight.

So what if you use the area where the two studs/beams are and use that as the window frame or opening, what if it wasn't a window, but instead a shelf? with a barn door feature interior of the office, you could close if you need sound barrier... idk i just had all these thoughts at once lol.

To enable the project reach a point where it might be usable but stay on a budget, you might consider focusing only on ensuring you have a solid pony/half wall and leave any glass or other part of the (upper half) of the wall until later. Alternately, postponing what you want done to ensure it will not "break the bank" to do it is a good idea.

Happy New Year.