Warning!
This answer is fuled by dark excel magic which is provided to you as a spreadsheet to look at here. I will not grant access to that sheet, but if you want to work with it, you can grab a copy for your own work.
Without Rerolls
To do this, you need to make a spreadsheet first: One of the dice goes 1 to 6 to the right, the other goes 1 to 6 down. This will fill out 36 squares with their sums. Or rather, an Array. For simplicity, we call the field in the top left [1,1] and the top right will be [1,6], bottom left is [6,1] and bottom right is [6,6].
Each of the fields has a number of 2 to 12, and each field stands for 1/36th chance. You can sum up all the fields (252) and divide by the number of fields (36) and get the average: perfectly 7!

From here, we can adjust to the other methods, and using a spreadsheet program, things get easy (if you know your black excel magic)
Replacing any outcome of <6 with 6
If the sum of 2d6 is replaced for any outcome below 6 with a 6, then all outcomes in the triangle [1,1] to [1,5] to [5,1] (or where the sum of the array indices is lower than 6) are replaced with 6. This leads to the top left corner being all 6s and an average damage of 7,55. The distribution of such a modified roll is this
$$6 = \frac {15} {36}\ ;\ 7 = \frac 6 {36}\ ;\ 8=\frac 5 {36}\ ;\ 9= \frac 4 {36}\ ;\ 10= \frac 3 {36}\ ;\ 11= \frac 2 {36}\ ;\ 12= \frac 1 {36}$$

If one of the 2d6 gets replaced
If only the lower of the two dice will be replaced by 6, the effect is much more powerful, the average damage gets boosted to 10.47. The distribution is now $$7 = \frac 1 {36}\ ;\ 8=\frac 3 {36}\ ;\ 9= \frac 5 {36}\ ;\ 10= \frac 7 {36}\ ;\ 11= \frac 8 {36}\ ;\ 12= \frac {11} {36}$$ Or in math terms: 7 has a chance of 2.78 %, 8 has a chance of 8.33 %, 9 has a chance of 13.89 %, 10 has a chance of 19.44 %, 11 has a chance of 25.00 % and 12 has a chance of 30.56 %.

Great Weapon Fighting...
Grand Weapon Fighting increases the average to 8.33 and thus by 1.33.

Replacing the Combined Dice
To make the table compatible with Great Weapon Fighting, you need to employ sub-tables.
Fields [1,1], [1,2], [2,1] and [2,2] are filled with the average of the whole table we already had, in the graphics in orange - so the value there is \$\frac{272}{36}\$
Fields [1,3] to [2,6] and its swapped counterparts need simulated rerolls: the 1 could be 1 to 6 after all. We can simulate that as the average outcome of the static die plus a roll of 1 to 6 (average 3.5), so the fields are as follows: $$[1,3],[3,1],[2,3],[3,2]=\frac{42}{6}\ ;\ [1,4],[4,1],[2,4],[4,2]=\frac{46}{6}$$ $$[1,5],[5,1],[2,5],[5,2]=\frac{51}{6}\ ;\ [1,6],[6,1],[2,6],[6,2]=\frac{57}{6}$$
The average damage output boosts to \$\frac{304.89}{36}=8.4691\$, an increase of 0.9136 compared to the no-reroll variant.

Replacing one of the 2d6
The same is true for the other method, we in effect need to have subcells for each rerolled chunk. Just the sum chunks are slichtly different. The average damage is \$\frac{394.56}{36}=10.96\$ - an increase of only about 0.488 compared to the no-reroll variant.
