Hey! Hopefully this advice can help you.
1) They don't look different at a glance-- They have an aura like everyone else. However, one of the things you can tell about someone if you take the time to read their aura (pg 313, core rulebook) is whether they are Awakened or not, and possibly what their "magical classification" is, which probably includes if they are an adept, magician, mystic adept, etc.
2) Astral Combat is the skill of fighting using your astral form, and so can only be used when someone is dual-natured (like magical critters and the Infected) or astrally-active (like a character using Astral Perception). It also is used when a character is only on the astral plane (like an astrally projecting magician, or a spirit that hasn't materialized). For your adept to use Astral Combat, they would need to have the Astral Perception power toggled on (or some other way to astrally perceive, like by smoking Deepweed). Otherwise, the character cannot be harmed by purely-astral beings, same as anyone else. Same, you cannot use astral combat to hurt mundanes or anything like that.
3) The Mystic Armor power functions entirely like normal, worn armor, except in its ability to protect you during Astral Combat (which normal, physical armor cannot) and that it stacks with all other armor with no penalty. Killing Hands has no special interaction with it.
4) Adepts aren't dual-natured-- Dual-natured refers to a specific type of creatures who are innately magical (metahumans aren't) and who are basically always astrally perceiving. Adepts and their powers may pass through a mana barrier without issue, but if you are astrally perceiving, this makes you astrally-active, and so means the barrier would stop you. So, just don't do that.
5) You're right. Unarmed Combat does not apply at all, unless you are a critter with the Natural Weapon critter power. Your complaint is also valid; I'm hoping to address it in errata. However, right now, a character's ability to fight in the astral is in no way affected by their other combat skills.
6) No; you're only perceiving the astral world. You have no extra ability to affect it during that time, beyond what you can already do. A magician could cast a spell to telepathically speak to their comrades, but that's something they'd just be able to do any time once they learned the spell. Additionally, you cannot see AR while astrally perceiving (AR does not exist on the astral, technically it isn't "on" the material plane either) and so would need to switch back to type of a text message to your team, at least.
7) Shock Frills are likely the same as Shock Gloves (only, obviously you don't need gloves). Their damage would then be 8S(e) with a flat -5 AP (as mundane electric damage has). You can't use them "defensively" if by that you mean "can I make it so people striking me get shocked". If you want that kind of thing, check out Shock Weave in Run & Gun on page 84. It also does 8S(e) with -5 AP.
As a note, you probably should avoid planning for fighting astral-only entities anyways. If there's an immaterial spirit harassing you, astrally perceiving will only allow it to make hit-and-run attacks on you-- It moves up, strikes, then flies away at the speed of thought. Unless you invest in some ability to counter-attack (which is possible) then you're a sitting duck. Otherwise, you still have the potential threat of immaterial spirits using mana spells on you (as Mana type spells can be can on astral targets while in the astral). If a spirit starts Stunbolting you from 50 feet in the air, your best option is to just tell it to suck it and stop astrally perceiving. You want those type of opponents to be forced to face you in the real world anyways, where you are stronger, even if you can fight them on the astral.