Pixels, inches and DPI
A print at 300 DPI needs 300 pixels for every inch. A 4-inch-wide logo is therefore 1200 px wide. The DPI value stored in the TIFF tells layout software like InDesign or Word what physical size to place it at.
Colour
The TIFF is RGB with an alpha channel. If your printer needs CMYK, convert in the print software or ask the shop; browsers cannot produce CMYK.