Analysis.

(Remember: this analysis is particularly true about British language and culture, though mostly it is the same for other English-speaking countries like Australia and the United States.)

Sentence by Sentence analysis.

Magnus Blake: "Hello Peter, how's it going?"
(This is a standard friendly greeting.)

Peter: "Oh, hello Mr Blake, fancy meeting you here."
(Peter pretends that he was not expecting to meet Mr Blake.)

Magnus Blake: "Yes, fancy."
C. (This reply is slightly sarcastic, and a bit unfriendly. Mr Blake is often in this bar, and he suspects that Peter is looking for him. Peter's pretence that this meeting is accidental is a slight insult to Mr Blake's intelligence. But he wants to know why Peter wants to see him.)
Your answer was:

Peter: "er .. Can I get you a drink?"

Magnus Blake: "No, let me. Barman - Scotch for me, and .. what will it be, Peter?"
B. (Usually the senior person will get the first drink, especially if they work together.)
Your answer was:

Peter: "A half of lager, please."
(Peter has chosen this drink because he doesn't want the boss to think he drinks spirits at lunchtime. Also some people regard lager as more sophisticated than the usual British beer, (called "bitter"). The "half (a pint)" is to show he is a moderate drinker, and anyway, Peter wants to be thinking very clearly.)

Magnus Blake: "So tell me, Peter, how are things going in your department. Everything ticking over ok?"
(Mr Blake suspects Peter wants to discuss something about work.)

Peter: "Oh, everything is fine, I guess."

Magnus Blake: "What's bothering you?"
(Peter has used "I guess", which here is a signal that everything is not fine, and Mr Blake should ask more.)

Peter: "Well, I wouldn't want to tell tales out of school ... it's nothing really."
C. (Peter is still pretending that he met Mr Blake accidentally, and he only spoke to Mr Blake because Mr Blake insisted. But "to tell tales out of school" means he has information which he thinks will interest Mr Blake.)
Your answer was:

Magnus Blake: "Oh, come on man, out with it."
(Mr Blake is less interested in playing games. Also now Peter has been given a direct order, he can say that he had to do as he was told if other people in the office ask him what happened.)

Peter: "Well, I don't really like to say, but the new boy, Ivan English, well, it's not really my job, but I do wonder, really, how well he's coping with building his new portfolio."
(Notice the over-use of verbal nulls "Well" and "really" to show hesitation and embarrassment. Notice the emphaiser "do" in "I do wonder". This suggests that really, Peter knows the answer very well.)

Magnus Blake: "Is he having problems?"

Peter: "Well, he comes to me and Sally for help all the time, so I suppose he must be. And when he makes a wrong decision, he's really arrogant about it. Just won't back down."
(Peter now gives his reason for wanting to talk to Mr Blake - he wants to say that Ivan can't do his job.)

Magnus Blake: "Go on."
(This is a completely neutral response.)

Peter: "I'm just wondering if he's really the right man for the job. I try to help him, since I have more experience, and I know the market better, but he won't listen. Still, I'm sure it will be ok."
(By mentioning his greater experience and knowledge of the market, Peter is suggesting he should take over from Ivan.)

Summary

Peter wants to pretend that he met Mr Blake accidentally, and his doubts about Ivan came up by chance in a conversation about work. This way he can pretend that he did not intend anything bad, but things "just happened". Mr Blake, however, knows that Peter has come to him to say that Ivan is not good at his job and should be replaced. Peter is being careful to say that Ivan always asks him for help (so that he can get credit for what Ivan has done well); but Ivan won't listen when Peter says he has made a mistake (so that Peter has no blame when Ivan has done badly). Peter also tries to put the thought into Mr Blake's head that he, Peter, can do a better job than Ivan.