Okay, it does indeed sound like these are actual DVD files and I would expect your player to be able to handle DVD+R (this seems to be the most compatible format). And don't worry about the AUDIO_TS; I've dealt with hundreds of DVD's and I have never seen one with any files in that folder.
I'm going to have to assume that the problem lies with the burning process. For starters, DVD Shrink isn't used for burning DVD's; it is only used for compressing them, putting them on your computer, and tweaking the files/menus. DVD Fab is essentially the same thing.
Now, many people swear by Nero, but I personally think it's a piece of junk. I've encountered many problems with it. So, to start, let's try a much simpler program (the one I use for all of my DVD's). Download a free trial of CopyToDVD here:
http://www.vso-software.fr/products/copytodvd/copytodvd.phpThe trial is limited, but it'll be good enough to test this out, and you'll be able to use it long enough to burn your videos. Install the program and when you start it, you should be given a few options. If so, choose DVD Video. You should then be given a dialog box that wants you to select VIDEO_TS.IFO. Navigate to one of your wedding videos, open the video folder, and then open the VIDEO_TS file. Then, near the top-right corner, click on the little icon that looks like a DVD on fire. Let the program burn the DVD and when it has completed (it will automatically pop out), test it in your player.
If it doesn't work, then I would suggest contacting the woman who copied these onto your computer for you. It's possible that she didn't copy them properly. Out of curiosity...open one of your copied videos. Do you see a VIDEO_TS.IFO file? (If it's associated with a program, it may look different than the other icons.) If so, double-click on it. Does your video play?