I’m not 100% sure here (others will hopefully clarify if not), but to try to answer why it’s not 会って来ました, when you chain verbs together using the て-form like this it usually means that one thing happened first and then the other after it (often because of it).
You can, for instance, say 出てきました if something (like, say, a pokémon!) appeared and came towards you.
Other examples might be 朝ごはんを作って食べました / Made breakfast and then ate it, or 悲しい映画を見て泣いた / Saw sad movie and cried (because it was sad).
〜ていく and 〜てくる can also be used to show that something is “coming” or “going” in time. As in 〜ていく is something that’s happening from now towards the future and 〜てくる is something that’s been happening in the past towards the present.
More on the last point here: http://www.guidetojapanese.org/learn/grammar/teform, way down under the header “Using motion verbs with the te-form”
EDIT: Looks like I was indeed a bit off… please take a look at the following replies.