A:

1. See the definition of "Stopped," above and "Dilemmas," below.
2. Yes, as long as it is made up of Personnel who aren't already "stopped" for that turn. For example, if your initial Away Team hits a Sarjenka dilemma and is stopped, you might bring up another ship, beam down another Away Team, and try the mission with them.
3. Yes. If you beam them both down it is assumed they are intended to be there together. If you want to operate them separately, first beam down one, try the mission (or battle, etc.), and if they get stopped then beam down another team, etc. The only exception is the case when one player beams down Away Teams from different affiliations and no treaty is in effect that allows them to be intermixed. In that case they are separate Away Teams in the same place.
4. If personnel were beamed down from multiple ships, then any dilemmas they encounter that affect the ship would affect all ships they beamed from.
5. Yes. You only need the proper affiliation to attempt the mission. Anyone can beam down to a planet, even if their affiliation doesn't match that of the mission.

A:

1. Yes. The rules say, (page 21) "The ships must share the same mission location and be owned by the same player." No distinction was made for non-aligneds between ships of different affiliations.
2. Yes, if your ship is going to be destroyed, you can try to save the crew by beaming them down to a planet first, where they will be stranded but can be picked up later. (They would be vulnerable to attack there from enemy Away Teams). Note that you can't do this is in the middle of a ship-to-ship battle because that is an action (see Definitions, above). Emergency Transporter Armbands would allow such a means of escape even during a battle, because the card specifies that it may do so. Note that a crew can also escape by using an Escape Pod card.
3. You can beam down the entire crew. It is assumed that there are certain "invisible crew" who perform mundane functions like beaming. After all, you don't really think that three people can fly the Enterprise, do you?
4. See "Missons," below.
5. No. These cards are a form of beaming and do not allow you to overcome any normal obstacles to beaming.