PNG is a raster format (like JPG). Raster images are made of colored pixels. When you zoom in, and the edge looks jagged, it's because you are seeing the actual pixels. When using raster images, you must not ever zoom or scale it, or else it will become pixelated. If you're in some situation where you expect the image will be zoomed or scaled, you should consider using a vector format, which does not have any pixels.
For an image on a letterhead, I would not expect it to be zoomed or scaled. Instead, I would expect it to always stay the same size. Are you in some situation where it will be zoomed or scaled, in regular use?
To export the PNG at the same size you create it, use 96 dpi, in the Export PNG dialog. If you make the Dpi larger, the image will get larger. If your requirement for the image is that it has a high resolution, then you should export at your required DPI. But then when you use the image, whether you print, or put it on a webpage, you need to specify the size you need, either in the print dialog, or the html on the webpage. That way, the pixels gets compressed down, and the image is the right size, with the higher dpi intact.
For making a letterhead, I'm not sure if you especially need a high resolution image. Have you seen, or are you following instructions somewhere, that tell you to do that? Otherwise, for making a letterhead, I would say that dpi is irrelevant.
Just make the logo the size you need in Inkscape, and then export at 96 dpi. Also, you should make sure Inkscape Preferences > Bitmaps > Export is set for 96 dpi.
I'm not all that familiar with Word, but I wonder if you could use a vector image?? You could try saving the logo as SVG, and try importing it. (By the way, you should be saving an SVG version of the logo anyway, in case you ever want to edit in the future. If you don't save an SVG version, you won't be able to edit in Inkscape - you'll have to draw it all over again.)
