When you have a spirit summoned, but it's not actually doing anything, it sits in astral space on standby, taking no actions. It takes a Simple Action to call a previously-summoned spirit (this makes it appear in the summoner's vicinity in astral space), then a second Simple Action to issue a command to a group of spirits.
As soon as it is called in combat, a spirit is entitled to roll its initiative; astral spirits all use (Force × 2) + 3D6. Since the Combat Turn is already underway, the spirit will need to subtract 10 from the result for each Initiative Pass that has already gone by. Note that while spirits have Edge equal to half their Force, summoned and bound spirits do not (SR5, p. 304); however, the summoner can use his own Edge on his spirits' tests, and they can use that to Blitz.
When the spirit is ordered into combat, what it does depends on its target. If the target is in astral space or is dual-natured, it can attack without materializing first. In this situation, the spirit must use Mana-type powers (like Confusion or Fear).
If the target is only in the physical world, or the spirit wants to use Physical powers (like Accident or Elemental Attack), it must first take a Complex Action to use the Materialization power. When it does, this changes its physical attributes to the numbers listed for its type (for example, a Force 4 air spirit changes its Reaction from 4 to 8), and its Initiative changes as well.
Per the rules on Changing Initiative (SR5 p. 160), you apply the difference between the base Initiative scores, then add/subtract any changes in dice. To use the Force 4 air spirit as an example, its astral initiative was 8 + 3D6; its physical initiative is 12 + 2D6, so you would add 4 and then subtract 1D6.
To revisit your example:
- Initiative 26: The mage calls his spirit (let's say its that Force 4 air spirit who rolls, say, 19), then orders it into combat.
- Initiative 20: The mook acts.
- Initiative 19: The spirit materializes. It rolls 1D6, getting a 3, so its initiative goes up by 1. Its next action will be on 10 instead of 9.
- Initiative 16: The mage acts.
- Initiative 10: The spirit wins the tie (because its Reaction is now 8), so it attacks the mook. The mook then acts, if he's able.
- Initiative 6: The mage acts.
Basically: Not really, unless they're pretty sloppy and don't counter your Astral Tracking. Although spirits might not be as unstoppably dangerous as you think.
Any spirit influencing material matters will leave a material trace, just like any physical being - because they must materialize to affect the material world.
If a spirit wants to affect anything on the physical
plane, it has to materialize first (p. 314). It gets physical
attributes based on its type (Spirits, p. 303) and appears
as a solid, physical version of its astral form—it’s relatively
solid even if it doesn’t look solid, like a spirit whirlwind
or a water elemental.
The spirit is going to leave some kind of physical trace at the scene, and an astral impression wherever it travels. That will fade relatively quickly though, and the mage can wipe it clean - so you're right, it is quite a challenge to track a summoner who wants to stay hidden when you arrive on the scene way too late. Just as it would be hard to track down a Rigger who used a sniper drone, or to find a decker who stole your files, or to find a sneaky assassin, 8 hours after the event when they've covered their tracks.
You do, however have Astral Tracking to fall back on if the mage hasn't erased their tracks everywhere the spirit has been.
Furthermore it seems to me that the astral signature vanishes with the spirit.
I'm not seeing where that's the case in the rules. You've even got a "+1 for every hour since the link was active" mod in the tracking threshold table (p315), which indicates you can track stuff that happened in the past after the link is gone as well as things that are happening now. If a spirit was there 4 hours ago, for example, this would suggest you can trace its origin at a +4 threshold (and tracking master by spirit is a +0 modifier for bound spirits, or +2 if it wasn't bound).
On the theme of danger...
A nonawakened person is pretty much helpless against a spirit
This isn't necessarily the case.
The materialization thing means that even unawakened victims can defend themselves quite effectively. Once the spirit materializes to make its assault they can physically attack it just like any metahuman opponent. Spirits do have all manner of freaky powers, and can more easily sneak up on the unawakened, but they still go down if you pump them full of bullets. Even before they materialize you've got a chance of spotting them:
Physical beings may sense when an astral form passes
through their aura. Make a Perception + Intuition
(4) [Mental] Test—you receive a +2 dice pool bonus if
you’re Awakened. If the test is successful, the character
feels slightly breathless, and they experience a chill or
tingling sensation from the passing of the astral form.
Security personnel are trained to recognize this feeling
as a sign of an astral intruder.
Dangerous, yes, but not really any more so than other sneaky backstabbers who might try to kill you.
Best Answer
The answer to this question depends on whether you are using Street Grimoire or not. As you note, in the Core Rulebook, the spirit selection is merely a guide, but the individual practitioner is able to choose whichever spirit the feel is most appropriate.
However, in Street Grimoire it is stated more explicitly that the summoner cannot task a spirit with a task outside its domain. From Street Grimoire, Second Printing, Pg 41, first paragraph:
No errata to date has clarified the issue one way or the other, but it is generally accepted (at least in my experience, and I think the devs had mentioned it as well, but don't quote me on that) that rules in supplements override rules in the core book when those supplements are used, on the assumption that they are replacing the appropriate sections with more detailed options, as well as being "newer". In other words, the supplements are expected to tweak some rules in the core book in order to allow the modifications and extensions they put forth.
So, if your table chooses to use SG, then that change (or clarification) is part of the package, though your GM can always choose to ignore it - most GMs I played with did, as it was overly restrictive and lead to some weird situations. For example, in the example in the book, the Earth spirit used by the Buddhist tradition for healing has no healing powers, optional powers, or skills. Of course, aside from Spirit of Man and their Innate Spell optional power, or Task Spirits and their Skill optional power, no spirit has much healing ability, making that a problem for many traditions even without the additional restriction, but almost all of them with it.
But as you've noticed, often one spirit is straight up better than another for certain tasks. Say you are fighting underwater, and your combat spirit is Fire. Even though you could summon a Water spirit, you are stuck with Fire because of your tradition. It's hard to imagine that this is intended, especially for a tradition like Chaos Magic, but RAW, it is the case.