1. I'd make this call according to the significance of the situation. If you're running the climax of your story, extended conflict resolution should be the norm - every action should count towards achieving something, but resolving it all in one roll would probably feel too short and simple.
Cases where a simple contested roll makes sense:
-Minor plot twists (you encounter an armed native of the planet who believes you're an invader from the neighbouring nation - how do you deal with him?)
-Inter-PC conflicts of minor significance (Two companions are arguing about which one of them knows The Doctor best, downtime at a UNIT base where two player characters are engaged in a game of chess and they want to know who wins)
-Conflict of little or minor relevance to the going plot (One of the characters, being a tough guy, tries to intimidate a local into giving them some information about the weird things that have been going on)
Extended Conflict should be used when you're trying to up the dramatic tension
-The Big Reveal - when the alien/monster/whatever that's been causing things to happen is revealed, it shouldn't be defeated with a single, scathing remark. This calls for extended conflict.
-In Media Res - If you want to get a session started with some real action, it should involve a lot more than a couple of dice rolls. Give the players opportunities to notice things about what's going on as the initial conflict unfolds.
-Your Players Like Fighting - If you've got a group that's accustomed to long combat sequences from other RPGs, it's probably a good idea to humour them with an extended combat scene, whether it's blasting Daleks with high-powered 51st Century Beam Weaponry, or trying to defeat the champion of the locals in hand-to-hand conflict to prove the virtue of your arguments.
-The Climax - Often the climax coincides with The Big Reveal, but sometimes the players already know what they're up against, and have had some time to prepare and plan for what they're going to do (The Parting of Ways, The Age of Steel, Doomsday, The Poison Sky, Journey's End, etc etc). If you get the impression that the players are planning for something big (or they just outright tell you "This is gonna be big!"), then it should get plenty of "screen time" via Extended Conflict resolution.
-The players aren't satisfied with the outcome of a simple conflict, so they're still in conflict. Let the conflict continue until they're satisfied with the outcome.
As for #2, remember that this is Doctor Who and not "Fighting Aliens that want to shoot you dead"
The right words at the right time are vastly more powerful than the biggest gun, which is a running theme of the game.
Also, there's the matter of degrees of success, so we really have 6 outcomes to look at.
A) Wants to talk, B) Wants to fire
A rolls presence+convince, B rolls coordination+marksman
Scenario 1: A succeeds/B fails
("Yes, But...")
A manages to keep B from shooting at them. Based on what A's player chooses to say, determine a reasonable "But" for the situation. For example, if A simply shouts out "I come in peace!", B might not be fully persuaded,
but, rather than shooting A on sight, will offer terms of surrender. If this is acceptable to the player, then you can advance to the next scene. If not, the player might try something else (say, picking the locks on the handcuffs, or trying to attack the soldier after their apparent "surrender")
Scenario 2: A has a good success/B has a bad failure
("Yes")
A manages to keep B from shooting at them. With a few quick words and a flash of their psychic paper, B concludes that A is a civilian who has wandered into a Caution Zone, and advises their immediate departure.
Scenario 3: A has a fantastic success/B has a disastrous failure
("Yes, And...")
A manages to keep B from shooting them, because B now believes that A is from the special division of mercenaries that were sent for. B proceeds to offer A a full briefing on the current state of the border conflict.
Scenario 4: A fails/B succeeds
("No, But...")
A us unsuccessful in convincing B not to shoot, but at least distracted B enough that his aim wasn't as good as normal. B fires a shot at A, but the damage from the attack is halved. (Note that if A's player is willing to spend some story points, the outcome may be further revised, such as causing the shots to miss)
Scenario 5: A has a bad failure/B has a good success
("No")
A is completely unsuccessful in convincing B not to shoot. Without caring about what A has to say, B shoots at A, and A takes full damage. (Again, story points can be spent)
Scenario 6: A has a disastrous failure/B has a fantastic success
("No, and")
A's words anger B significantly, who rather than firing a single shot proceeds to empty his firearm in the process of shooting A. A takes half again as much damage as normal. (Story points can, and probably ought to, be spent)
The right words at the right time are vastly more powerful than the biggest gun, which is a running theme of the game.