I think it's easiest to make a table of the responses you want first, and then look for the questions that get those responses. The result of the first question (WLOG assume it is to A):
ABC
===
TFR da
FTR da
TRF ja
FRT ja
RTF __
RFT __
Like you said, if we get da we know B is not R, and likewise ja means C is not R. So we can ensure our next questions do not have any random (__) responses. Then you can easily find a pattern that identifies every god
ABC
===
TFR da da da
FTR da ja ja
TRF ja da da
FRT ja ja ja
RTF __ da ja
RFT __ ja da
You can now mechanically come up with appropriate questions that produce each column.
> whether this puzzle is solvable by asking a question to god A, B, and C in succession
It's not. The 8 responses have to be mapped onto the 6 permutations of ABC. If A, B, C are asked in succession, one question will always be to R, and which permutation we map to cannot depend on the answer to that question. For example, if ja ja ja maps to TFR, ja ja da must too. That means for each of the 6 permutations, at least 2 responses must map to it. But that would require 12 possible responses.
By the same argument, only the first question can be to R. That means the solution is basically unique: use the first question to make sure the next questions are not to R.
> whether this puzzle is solvable by asking a question to god A, B, and C in succession
It's not. The 8 responses have to be mapped onto the 6 permutations of ABC. If A, B, C are asked in succession, one question will always be to R, and which permutation we map to cannot depend on the answer to that question. For example, if ja ja ja maps to TFR, ja ja da must too. That means for each of the 6 permutations, at least 2 responses must map to it. But that would require 12 possible responses.
By the same argument, only the first question can be to R. That means the solution is basically unique: use the first question to make sure the next questions are not to R.