Yes, you can put a perfect construction after since, but the first example seems slightly clumsy.
Normally you would say:
It's been a long time since I last saw your face.
It might be an attempt to be artsy, but unlike most violations of grammar in poetry or song, this doesn't sound deliberate to me. I think it's just a sloppy but acceptable analogy with constructions like these:
I've been lonely since you've been gone.
You haven't been late even once since I've been working here.
The perfect present means that some time interval ending in the present contains the action of the verb. (That's not a settled, scholarly opinion, but it's what I favor at the moment. More here.) Using the perfect present twice like this means that the same time interval contains all the "working here" and contains no "being late". The continuous aspect after since indicates that "gone" and "working here" extend continuously, without a break, filling all the time since they began up to the present.
In the example, "seeing your face" is an event at one point in time, not an action that runs continuously up to the present. "I've seen your face" means that the speaker's experience up to the present includes seeing your face, without referring to the specific time when it happened. But the example tries to use "I've seen your face" to establish the specific time, which serves as the start of the "long time coming". It doesn't quite agree with ordinary usage. In ordinary usage, you use the simple past for that.
Probably the authors were trying to emphasize the unbroken continuity of the period of not seeing your face, and just chose a weak way of doing it. The continuous aspect in "It's been a long time coming" agrees with that. If I change "I've seen your face" to "I saw your face", it actually sounds a little more jarring, because "It's been a long time coming" seems to call for another perfect construction to explain what "it" is and what it's been doing during that long time. But the sentence as written is like "I've been lonely since you've walked out the door." "Walked out the door" is a single event, so the perfect present doesn't quite work there. A listener's mind can bend the usual grammar enough to handle it, just with a lingering feeling of sloppiness whose origin is hard to identify. (Maybe that feeling is why you posted your question!)
P.S. I checked out the song and found that in the next verse, it stresses the final syllables of "working" and "nothing" in order to force a rhyme. It doesn't appear to be a joke, the way the same thing is done in this song. So, I think the slight abuse of the perfect tense resulted from general carelessness on the part of the people who wrote it. It's in the fringe of sloppy usage that's not so serious as to attract much notice--except from someone like you, who's paying close attention.
Both are grammatically correct.
In distinguishing between them, You would "typically" use the first example for a single event, and the second example for a recurring series of events.
For instance, if you want to schedule a meeting with your boss to discuss something specific, you would use the future tense. ("I will be free . . .")
On the other hand, if you are a professor who has posted weekly hours, you would use the present tense. ("My hours are . . .")
However, there is no rule that says you have to use one or the other. It's just a matter of convention and preference.
Best Answer
The first three aren't grammatical for these reasons:
the structure X then Y requires X and Y to be events. The phrase "It was only ten years ago" isn't an event; and
the structure X since/after Y requires X to be a duration (e.g. ten years). The phrase ten years ago is a specific point in time, not a duration.