If you are comfortable with photoshop, yes, you can just just change the color of the image files in the psd files and then upload the new files. I see no issues there.
For the stylesheet changes, you can copy the content of the style.css file or any other css file you changed to the child theme style.css file without causing trouble. If you are still concerned about losing changes, you can use either use DiffMerge(Mac) or WinMerge (in Windows) tool to compare the contents of your server with that of the original contents and figure out exactly what you changed and move them over to the child theme.
Any issues with logo on the mobile, if you are checking this on your mobile, pls make sure you have set the retina logo under Logo Options in Theme Options. You can also take a look at this post for another solution if you do not want to set a separate retina logo for your site or just need a temporary solution –
https://www.livemeshthemes.com/support/forums/topic/google-chrome-problems/#post-3293