What is the proper amount of time to thaw + brine a turkey simultaneously?

A few things aren't quite right here.

1- a cooler full of water for 3 days.
A turkey will take 3-4 days to thaw in the fridge- between 35-40F. Quicker methods call for submerging in running water for some hours. Submerging the turkey in stagnant water- even if it started as ice water- will allow the turkey to rise well above 40F over the course of 3 days. You would have to add ice or have the turkey somewhere colder than 40F for this to be safe.

2- brining too long turning a turkey to mush.
The salt in a brine denatures some proteins and supercharges the bird with water. These will make the meat more tender and seem more tender respectively. That said- the risk isn't that the meat will get mushy. Unless you are adding some other ingredient to the brine that is actually a proteinase like papain the risk is not mushiness but being too salty.

3- 1-2 days of brining.
I have never seen a recipe that called for brining this long. The brine recipes I have seen are on the order of 4-10 hours. If your brine recipe calls for this length of time then it is probably more dilute than the ones I use. In such a case it would probably be fine to thaw it in the brine if you address the safety concern.

The turkey being frozen will prevent the brine from penetrating but as the bird thaws from the outside in the outside of the bird will have more alone time with the brine than the deeper meat. I can't say whether this would be a problem- especially as Harold McGee says that the salt from brining isn't able to penetrate very far into the meat anyway.

Unless it is cold enough outside (or you have enough fridge space) that you could brine the turkey at <40F for 2 days while it thawed I would recommend at least mostly thawing it before introducing it to your brine.

Alton Brown was on NPR's "All Things Considered" this week.

http://www.npr.org/2012/11/13/165039668/turkey-tips-from-alton-brown-dont-baste-or-stuff

He said he likes to thaw the bird in the cooler for up to a week in ice brine. As the ice melts it dilutes the brine. I am trying this myself this year, but only for 3 days. Alton claims he has not seen the water get above 38 - 40 degrees when doing this, however; I would put some common sense to this and assume it would all depend on where you keep the cooler. I live in Fla. so I will keep it inside in the A/C and just add ice and drain a little brine, if needed, as the thermometer drops.

I have used Alton's turkey cooking method of high heat (500) for first 30 Mins and then 350 till 161 degrees, and it was the best turkey ever. My 72 year old Uncle-in-law, said it was hands down the moistest he has ever had. Good luck!!!

This was the second year I brined a turkey and I accomplished the task in a bucket with brine and ice + water, just as I had the year before. I did this for two days this year because of a last minute issue that arose on the day I had planned to roast, pushing my roasting day back by 24 hours. I was really worried my bird was ruined, but not having a choice in the matter, I went ahead an opted to roast it and see how it turned out. I was not disappointed in the least and it was as wonderful as I remembered from the year before. My bucket sat out in my breezeway this year for two days (I live in PA)and still had plenty of ice in it when I went to retrieve it. This is hands down the most wonderfully moist turkey I have ever had, made, or had compared to anyone else's! Even my boyfriend told his mom in front of me that mine turned out better than the one she made two days earlier ( I wanted to crawl under the table! ). Try it.... You will never prepare a turkey any other way. (Maybe change the brine recipe, but) you won't use any other prep method again.

You could also consider equilibrium brining. Since you sound like you want to leave it to thaw for several days, this may be a safer way to avoid over salting the meat. The basis of the idea is you want to weigh your bird, decide how much salt you would want in the final product (i.e. maybe 1% salinity) calculate that weight in salt (bird weight X salinity %) and make your brine that way. This way the % of salt in the brine is at the level you want in the bird and therefore the amount of time you leave it in the brine is irrelevant.

This is a method that was invented by the team behind Modernist Cuisine. There is some info out there if you google it but to really understand it your going to have to spring for the book which is about the cost of 5 thanksgivings ($516 at the moment)

http://amzn.com/0982761007

You cannot thaw a turkey in a brine. The cell walls are frozen solid and no saline solution can penetrate those type of cells.

You can thaw a turkey much quicker by placing the bird in some sort of plastic bag and immersing it in lukewarm water but this is not entirely what you are proposing.

See this related answer. Can you brine frozen meat?