No. Your PPI (pixels per inch) does not change if you just cut the half of one image and paste it above the other half. The only thing it changes is your image dimensions. So your width or height becomes half based on how you decide to cut it.
If you want to change DPI of your image. Load it in Photoshop. Then go to Image > Image Size or press Alt + Ctrl + I. Go with your preferred values.
No. The PPI would depend on the resolution of the document/image in which you were adding or combining another document/image. If the original or recipient document/image is set to 72ppi and the document/image you were combining or donor documents/image ppi was higher or lower than the recipient, the recipient documents/images ppi would remain the same up and until you change it. The only thing that could change would be the dimensions of the document/image.
Even if you cut both documents/images to be combined in half, the receiving document/image will determine the ppi of the combined documents/images regardless of how either documents/images dimensions are altered.
Other than dimensions, the only thing subject to change will be the file size. Even if the recipients documents/image dimensions remain the same, adding or combining a new document/image, regardless of size and ppi, will increase the file size of the combined document/image.