As has already been pointed out in the comment above, your assumption about a resolution of 360 is incorrect. The necessary resolution is dependent on the final use of the image. A good rule of thumb is that the image resolution should be twice the screen frequency of the printing method. So a magazine printed at 175 line screen would require 350dpi images for optimum quality, while a carton printed at 85 line screen would only need the images to be 170dpi. Different rules apply to digital printing, but in either scenario the print company should be able to provide you with a specification for what is required.
Supplying files at 72dpi, but with large dimensions is common practise, but this varies from supplier to supplier and many of the more professional image sources provide images at 300dpi by default or at least offer that as an option. It doesn't matter which way the file is supplied because you can go to Image Size in Photoshop and adjust the size. Be sure to uncheck the Resample Image option so that the resolution increases as you reduce the dimensions. Scale the image to the size that you need, check that the resolution is high enough (see above) and if the resolution is way too high then so back to Image Size and reduce the resolution (this time with Resample Image checked).
The different file sizes that you are seeing are the compressed and uncompressed sizes. The value in Photoshop is the uncompressed size of the actual image data which is fixed for a given size, dpi and colour mode. The smaller number is the size of the file after compression and this will vary for different images and file formats. For example, the data for an A4 CMYK image at 300dpi will be roughly 32MB, but the file size will generally be MUCH less, depending on the compression method and amount of compression applied.
There are many resources for information on this topic on GDSE, just search for things like 'image size' and 'resolution' and keep reading and experimenting until it makes sense.