OK, I did most simple simulation just for yes/no for tribute and below are the plots. Horizontal axis Tribute shows number of kingdoms giving tribute at checking. Frequency shows how many times this result was encountered during testing and in simulation. As one can see, the plots are essentially identical.
Since the distribution of data is not normal, some non-parametric test should be used to compare the actual and simulated data. I used Mann-Whitney test but I’m sure some other test can be used although it would yield same results. So, P value is certainly not less than 0.01 or 0.05 or even 0.1, it is actually 0.9 for 2-tailed version. This means that both sets of numbers come from the same distribution and are essentially identical at very high probability, about 99.99% or maybe more 9s there, not sure how many. This, in turn, means that chance for tribute is random and independent and exactly matches in-game description. So, working as intended 100%, nobody is trying to deprive honest players of their hard earned goodies, certainly not the developers. 
Now, the other part with gold, glory, and souls is a bit more complex, I need to spend some more time on that.
And here is the table I used.
| N | Simulated | Actual |
|---|---|---|
| 1 | 0 | 0 |
| 2 | 5 | 5 |
| 3 | 7 | 11 |
| 4 | 25 | 19 |
| 5 | 57 | 46 |
| 6 | 75 | 73 |
| 7 | 100 | 100 |
| 8 | 111 | 129 |
| 9 | 104 | 101 |
| 10 | 90 | 85 |
| 11 | 55 | 51 |
| 12 | 32 | 35 |
| 13 | 20 | 21 |
| 14 | 8 | 14 |
| 15 | 5 | 4 |
| 16 | 4 | 3 |
| 17 | 0 | 0 |
Simulation data set is too big, I can upload but it is very simple set of equations. You create an array in Excel of =rand() and then create another array using IF function =IF(A1<0.26,1,0). And then just count the number of 1s in each row. Then, autofill frequencies and count these as well. All is quite trivial.
I got a simulation for gold and the sum total for 700 tributes almost matches the actual result. Actual total is about 3.3 million and I got 3.1 million. However, the sums don’t match well with souls and glory, there might be some error somewhere.