They are in the same Arthur packet. The A-packets are designed exactly to answer this question. I think the answers to questions 1,2,3 are "no", "yes conjecturally", and "no": see book of Adams Barbasch Vogan or Vogan's survey article on the local Langlands conjectures. Note that, although the A-packet is not a union of L-packets in general, it at least contains a canonical L-packet.

They are in the same Arthur packet. The A-packets are designed exactly to answer this question. I think the answers to questions 1,2,3 are "no", "yes conjecturally", and "no": see book of Adams Barbasch Vogan or Vogan's survey article on the local Langlands conjectures. Note that, although the A-packet is not a union of L-packets in general, it at least contains a canonical L-packet.

[Oops: answer edited. This only applies to "automorphic" in the sense of "appearing discretely in the L^2-spectrum". Misunderstood the focus of question. Explicit reference for 1 is e.g. 1.11 of Gan-Gurevich "Non-tempered A-packets of G_2: liftings from SL_2-tilde."]