So the Google Photos app isn't going to copy your photo library to your microSD card, it's backing up your photos to your online Google account. And it's not going to happen within minutes, as I wrote out previously if you have a large number of photos the backup process could take while. Depending on how many photos you have and the speed of your online connection it could take several hours. By using Google Photos instead of manually backing up your photo library to a microSD card, all you need to do with your next phone is install the Google Photos app to access your photos again, plus you have access to all your photos even if you don't have your phone with you.
As for your microSD card, that's another issue. Going by the error messages you've referred to, it sounds like the card needs to be formatted so your S5 can actually use it. You can do this by going to your Settings >> More >> Storage and scroll down to the bottom option 'Format SD card' Note that this will wipe everything that might be stored on card. If you have saved anything to that card previously, DON'T format the card. Take your S5 to a phone center/dealer or to a more tech savvy friend and tell them what's going on. If you do have things like photos or music stored on the card that data has to get backed up first, then format the card. If you've never used the card before and you haven't saved anything to it, then go ahead and format it. Unless the card itself has gone bad at that point your S5 should be able to recognize it properly, and that should take care of those '... invalid' messages from popping up.
Just my opinion but moving to a S7 instead of a J5 is much better option. The J5 would be more of a lateral move from your S5, the S7 on the other hand is an upgrade.