The other problem with Spirits is, they have never been very clear exactly what they can and can not do over the edition changes.
From 1 to 3, we have a pretty defined idea of what elementals (the Spirits that Hermetic mages summoned) and what Spirits (as what Shamans could summon). An elemental always had the personality of a Rock (No, Not the Wrestler!). You ask them a question, and they just blankly looked at you... (Non-sapient). While a Hearth Spirit could talk your ear off about that time Washington came by and dropped a flaggon of ale on some poor maid's foot, thus causing a near riot!
And then along came 4e... And they kinda of shoe horned elementals and Spirits and Mages and Shamans into one giant pot without really taking the time to tell us EXACTLY what and how Spirits function now...
Now, for my take on the above Question of "can a Spirit use Skills"....
the default answer is "yes. With exceptions".
The first thing to remember is that Spirits, even when materialized are NOT biological entities. Meaning they don't "See" with eyes. They see with Astral Perception only. (as poorly worded as it is in the CRB, and covered in the 4e book before it).... So this in itself is a HUGE obstacle to overcome...
Just imagine asking a blind person to do the said task. Makes "Press the Red Button!" actions kind of impossible huh?
The Next is that while Spirits are Sapient, and thus smart, they don't really understand the physical plane. (Remember, Spirits are not affected by stupid laws of phyics or gravity! Those things are for Mortals!) Couple that with half our world being a grey wash (anythinb not living) with spots of vibrant color (life and Emotion).. its not something that translate well into skills that a Spirit could use that would help a Shadowrunner...
Take "Lock picking" for example.
Well, if its a electronic lock, Then the Spirit can't even read the screen! Or tell one wire from an other to hotwire it...
If its a Combination lock, it can't read the numbers nor even see the arrow to align with...
If its a key lock, it might be able to figure out that a small piece of metal might be able to move the internal keys into the right position....
But, More likely it would be thinking: "Why"? If the portal on the grey wall won't open, just go through the wall and re-materialize. Or fly over the fence. Wait? you want to open the grey Portal, in the grey wall, to retrieve some grey items from inside??? Why???
As to summoning The same spirit over and over again to teach it a skill...
Well, Things get really murky here.... with 2 different lines of thought and 2 different pieces of supportive info for their cases.
Basically, it boils down to a combination of things:
The first is either Spirits share a Hive mind intelligence, and when you summon a spirit, you are calling forth a small sliver of this hive mind for a limited time... And this is the reason why actions with one Spirit will/can affect interactions with a different spirit later on... Because they are all one giant intelligence..(See: Spirit Rep)
The other line of thought is that Spirits kinda of act like a Girl's locker-room does for Rumors... but on a planetary scale. And thus each spirit is a separate entity, but it's through this "Spirit's locker-room" that other Spirits learn of your miss deeds to Spirit kind... (again: Spirit Rep)
Sadly, there is nothing concrete either way, as everything with Spirits is (probably intentionally) filled with contradictions. Spirts have shown knowledge of things that they shouldn't have knowledge of. And shown a lack of knowledge of what it should know on others... Spirits and tech don't work because biologically, Spirits are just blobs of Ectoplasim... yet there are accounts of Free Spirits BTLing out...
So yea... Basically, thanks to how little is codified, and the wording of what we do have... GM call to just about anything for a Spirit can do with skills... Or if it can use skills...
Hopefully your GM is up for the task..