Hi, anon chaaaan!!! ♡＾▽＾♡
First of all,please, be kind and forgive my delay!(。┰ω┰。)
Now…

-About the first question, I answered it few time ago..
Because my knowledge on the subject hasn’t changed or improved, also my answer wouldn’t change, so, I hope my anon won’t get angry if I redirect her/him here!!
-As far as concerne the second question, we have more canon infos, thanks the release of LSW and KMK!!!
(premise: I have not had the opportunity to see the movie yet, so the little I know I’ve learnt from some spoilers on tumblr and thanks to some chat with Japanese friends„„so…if someone wants to add or correct something, he/she will be welcome  (‘ε’)  )
The  first mention of green clan’s power is found in the first chapter of manga “K The First”:
 the green growth of “transformation/manipulation”
Actually it is a too general and too ambiguous sentence to comment
on it (at the moment)!
Basically, green clans is made up of ninja!
(Yeah, Fushimi-kun…I’m sorry but Munakata-san is always right!)
Not only because they wear ninja mask..or use ninja weapons, but because they totally practice Ninjutsu, the sublime art of stealth and espionage!!
Obviously, K isn’t set in Sengoku Age..thereforet they take advantage of new ways to get information: through the net!
they have the power of working through it and connect to a network and take/send several “info” (and not only)
“Jungle” is one of the app they use to take info!
They can also create openings through walls.
(oh I don’t know how can they do it, but they CAN DO IT!)
*update*
according a japanese friend of mine, They turn the body and walls into electronic, so they can pass through the wall….*cool*

We officially know ¾ members:

Nagare Hisui, Midori no Ou, The Green King, whom power is linked to electricity and electromagnetism….
in KMK, he showed his power making electric shock and a “Thunder Dragon”

Mishakuji Yukari: Nagare’s right hand, he was a former clansman of the previously Colourless king, Miwa Ichigen, and, of course, is Kuroh’s senpai, became afterwards an effective member of green clan!
he is fast and agile in fight and with his katana* gave quite a hard time to Kuroh!

Hirasaka Douhan: patient and analytical fighter!An excellent ability to fight and his sharp mind make her an amazing ninja!
(btw… she’s a cute blond-girl!! （*＾3＾） )

Kotosaka: He/It ‘s a parrot, used by Nagare-san as his own vessel!!
(in KMK)     

And about the 3rd question….believe me if I say I’m more shocked than you!!!
According to K-Side Red, a strain could never became a king…and, in Anna’s case, she would have become a king only (and maybe) by forcing the slate!!
But, in the movie, she did nothing like this!!
I hope Gora will also clarify this point!
(Or, if someone who watched the movie *or not* knows more, please, let me know!!!!)

At the moment, this is all I can tell you!!!
I’m sorry if my answer is not satisfactory,not clear or other!

HAVE A NICE DAY!! (｡･ω･｡)ﾉ♡