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### $E_{\pi}[R_{t+1}|S_t=s,A_t=a] = E[R_{t+1}|S_t=s,A_t=a]$?

Question: Can I write it without the subscript? So $$E_{\pi}[R_{t+1}|S_t=s,A_t=a] = E[R_{t+1}|S_t=s,A_t=a]$$ Yes, your reasoning is sound, there is no need to condition the expectation on the policy, ...
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Please look at line 5: If $P(a_{i,j}|s_i)$ is equal to the policy that is used for generating the demonstrated trajectories, then it could be the same. However, in inverse RL you don't know \$P(a_{i,j}|...