Out of the two,indole

3 acetic acid is more effective than 2, 4 D....

For hairy root transformation after co-cultivation you just go for MS+2,4 D (0.5 mg/ml) in less concentration or u can use IAA in 2-4 mg/ml.

I do hairy root transformation in vetch. I did not use auxin in the inoculation nor the culture after that.

As Pratik Pusadkar correctly suggests, you may use either IAA or 2,4 D, with particular attention to the concentrations: IAA must be applied at higher doses than 2,4 D, because it undergoes metabolization by plant tissues (catabolism and conjugation).

2,4 D is better than Indole 3 Acetic Acid for plant tissue culture in hairy root transformation. For thickening stem usually indole acetic acid is used.

Antibiotic is more preferred for soybean tissue culture is imentin than carbenicillin

It depends on the nature of the plants...IAA itself is an organic auxin, which is found effective on root induction of most plants. Perhaps 2,4-D is good to induce roots via indirect organogenesis when there is a callus stage in between the process.

I found that many papers used NAA instead, perhaps due to the heat-tolerant ability that is found under synthetic auxin.