As a rule of thumb, the perfective form refers to a completed action, while imperfective form implies that the action was in progress (and not necessarily finished). However, in real life there are a lot of nuances :-). Let's just look at your particular examples:
Example 1:"я видел его вчера" vs "я увидел его вчера"
The above-referenced rule of thumb doesn't apply here because the verb видеть already implies successful completion. You can't (пытаться) видеть, видеть, видеть, and finally увидеть. To indicate attempts, there's another verb - смотреть. So, in practice we use видеть instead of увидеть. Still, увидеть refers to the particular moment when it happened, and it changes your status at that moment — before, you didn't know something and afterwards, you know it. Like "я увидел его вчера с моей женой" vs "я видел его вчера с моей женой" - the latter is more neutral, whereas the former may imply that it was a discovery for you :-)
Example 2:"он звонил мне вчера" vs "он позвонил мне вчера"
Here the above rule of thumb is applicable - звонил may mean that he made my phone ring, but I didn't pick up the phone. So he called, perhaps several times, but with no success. In practice, when we say "он звонил мне вчера", we may mean that we actually talked (it wasn't necessarily an unsuccessful attempt), we just don't emphasize the success of the operation. For example, он звонил мне вчера, и мы хорошо поговорили. Here, звонил is used instead of позвонил because the distinction isn't so important; what's important in that statement is that вы хорошо поговорили. If somebody who hasn't called you for ages called you and you want to emphasize that you talked to them and it had an immediate effect on you (the fact that they called), you would say позвонил. For example, Президент компании позвонил мне и поблагодарил.
Example 3:"ты купил молоко?" vs "ты покупал молоко?"
Here the above rule of thumb works perfectly: покупал - means "was in process of buying," but maybe refused to complete the purchase in the end. Купил молоко - definitely means that some amount of milk changed the owner as a result of the transaction.
Example 4: "я уверен мы встретились раньше" Vs "я уверен мы встречались раньше"
As far as I'm concerned, "мы встречались раньше" is simply the more idiomatic way to say this. People get used to saying "мы встречались раньше". Only non-native speakers will commonly say "мы встретились раньше" in this meaning, as in your example, i.e. when you have a feeling of knowing somebody's face. However, you can use that form in other contexts, for example: Вы встретились с ним 25 февраля? Нет, мы встретились раньше.
Example 5: "они видели этот фильм 6 раз" - why did we not use "увидели"? Because, as already mentioned in another answer, увидели assumes some novelty. Увидели means that it was a discovery of some sort (it changed your state of mind). You can't discover the same thing six times.
One more thing worth mentioning here: In Russian, prefixes are used to create the perfective form of a verb, but these prefixes (aside from putting the verb in the perfective aspect) may add some additional meaning or connotation. For example, if you use the phrase "они смотрели этот фильм 6 раз" and "они посмотрели этот фильм 6 раз" - here both phrases are possible. "Посмотрели" is the perfective form, but it doesn't imply "discovery" as is the case with "увидели".
Example 6: "в последнее время я о ней не слышал" - why did we not use "услышал"?
Here, it's also almost idiomatic (as in the case of "встречались раньше"). Here, a general feeling is discussed: "I don't remember getting any news about her recently".
