Been there, done that
Chuck, several people here have dealt with flooding or major water leaks (including minor drips which ultimately are just as bad).
So let's hope mine is only one of many postings.
Here's the basic answer to your question:
Yes, it is doable and, yes, you can do it.
That said, here is how I have approached such problems.
First, figure out a second bathing solution for the next several weeks.
Second, you are going to have to rip and rip and rip until everything moldy and or wet is exposed. Wear a mask.
There are several excellent articles out there on treating mold, personally I suspect Lysol spray is just as good as any other, once things are dry.
You may have structural damage, tho' probably not. Given the weight and the fact that everything is hidden away, that section of the house is designed to carry a heavy load without shifting or settling. This is the only place I would be more comfortable having a professional come in and check. Your insurance adjuster can probably do it for free, after all, their main goal is to say, "sorry, not covered".
Once everything is dry and known to be structurally sound, your next step is to rebuild the three surfaces involved (obviously, the plumbing will have been fixed).
The floor is the easiest part, especially if you stick with standard American dimensions.
The ceiling most not be overlooked and you should at least consider ventilation, if possible. In any case, the ceiling needs to be designed without any cold corners or surfaces so that water doesn't condense in one section and start causing problems.
The walls can, now-a-days, be done with materials which are not only easy to work, but also meet fire codes, provide ideal backings for ceramic tile and are rot, insect and relatively mold-resistant.
What to do? First, check out all the helpful tips on the internet. There are several excellent video clips on each step.
Second, check out a good book on bathroom remodeling, preferably one from Better Homes and Gardens, they seem to have focused on normal people like us and not engineering graduates.
Third, when doing it, think about access panels. Here in Europe, we use Styrofoam holders for shower and bath basins. In the side, a square or oblong hole is cut the width and length of 4x4 or 2x4 tiles. Special plates which hold those tiles and mount with strong magnets are set in. I have seen them in the US, too, and they are well worth the minor extra expense - you no longer have to get everything just right before putting in the basin or the tub, but can do the project in steps. And, if something, make that when something, goes wrong, it is an easy fix.
Do not use plastic for water pipes under pressure, no matter what people tell you. Use copper. Do not use epoxy for the fittings, either sweat them or shark-bite them. Even if the gorgeous guy in the plumbing section of Homo Depot insists that only old fogies do CU, all the "real" pros use plastic. My darlin' got his basement soaked with real, NSF approved plastic not all that long ago. Was my fault, too, which didn't make things any better. I now use stainless steel braided neoprene for flexible connections and sweat copper in for the rest.
Do not save money on anything which will be constantly wet or hard to replace.
Don't go overboard on exotic patterns and designs. I did a black bathroom - a black tiles, fixtures, sanitary and pipe fittings for a couple back in the late 1980's. They loved it. When they sold their condo in the mid-90's, they lost 20,000DM value, 'cause the market had switched to the colors of autumn.
Today, of course, the bathroom would be en vogue.
Hope this helps. I tried not to go technical, but I have the experience and have made enough mistakes to be able to answer more detailed questions. Some, have serious professional experience. Just keep at it.
Oh, and, good luck. It sucks, but at least no one was hurt.
And why did we get the picture only after you were out of the shower?
