Here's the deal on the pronunciation of the first k in arkitekt: both a hard [k] sound and the [ɕ], as in köper or Kina are acceptable.
The pronunciation with a k sound is more prestigious and is preferred by most actual architects, and it's also placed first in SAOL, which means it is the one they recommend. However many (probably most) native speakers, myself included, prefer the soft [ɕ] sound.
The k pronunciation goes back to the Greek pronunciation, whereas the other one is because we loaned the word via French and German.
No, it's "Have you rung the architect?".
Ring, rang, rung.
Rung is the participle. "Have you rung" is like "Have you went" — dialectal.
And yes, obviously, it goes without saying that you can ring someone up instead of calling them up. All speakers of standard English will understand that. Americans happen not to say it like that, and Duolingo has a huge US bias, so we're in the weird situation of translating ringe as "call".