Learn English – I ….. him since we met a year ago

Since in this context means after (i.e. - after some time explicitly or implicitly identified in supporting text). It might help OP to imagine replacing since with after the time at which in both his examples.

In other contexts, since can mean because, but that's really just a metaphoric extension of the above.

I think what's bothering OP is that since has many subtly different usages. Here are just a few from OED:

1: During the period between (a specified time) and now; at some time subsequent to or after.
2: Ever or continuously from (a specified time, etc.) till now.
3: Denoting a point of time to which the action or event mentioned is subsequent.
4: In sentences implying continuity of action or fact during the period indicated. Also with ever, and (rarely) with that.

Most (but not all) senses imply continuity from whatever time in the past is indicated by since, up to now. For example, "I have known John since we were children" is a very common context, and we normally use past perfect there because we still know him now.

But suppose you were at John's funeral? It would be perfectly grammatical to say "I knew John since we were children" (you don't really know him any more, since he's dead). I suspect many native speakers might have the same misgivings as OP, but I think it's just because we're more used to contexts where there is continuity into the present. FWIW, I think I might be tempted to say I knew him since when, or even since way back when, to distance myself from the standard up till now/ever since then implication.

I haven't seen him

means, essentially, I have not seen him recently.

I didn't see him

means, I did not see him at some particular time.

When the "particular time" is recently, the two can be used almost interchangeably:

"Have you seen Bob around?"
"No, I haven't seen him."

"Did you see Bob just walk by here?"
"No, I didn't see him."

Other times, though, there are clearer dividing lines between when to use didn't see instead of haven't seen:

"Have you seen Mr. Hernandez go into Mrs. Olson's house?"
"Yes, I have seen that many times."

"On the night of the murder, did you see Mr. Hernandez go into Mrs. Olson's house?"
"Yes, I did see that."

The first question refers to a more open-ended span of time, while the second refers to a very specific time segment.