Requiem is indeed too mannered here.
Of those three you mention, I might go for Maiola.

hhp

I think either Maiola or Scala have the right feel for what you are doing. Maiola gets my vote - it is a solid choice.

FF Quaadrat is a beautiful face, but it is so special, it tends to grab attention rather than blend in. I purchased an art book (on Louise Bourgeois) a while back typeset in FF Quaadrat. It is written in German, which I don’t read, but the look was so striking, I just had to have it. It looks marvellous on the slightly yellow paper and it fits perfectly with the (contemporary) topic. I tried to typeset another book myself – a photographic documentary book – where it didn’t work as well. To my eyes, it turned out too expressive. A little bit too much. Fred Smeijers is IMO on of the finest type designers of our time. You should definitely have a look at his other typefaces. For your project, I would probably consider Arnhem, Custodia (if not for the strange q) or TEFF Renard (if not for the steep price) before Quaadrat. But Quaadrat is classy, just super-edgy-classy.

Maiola, by Veronika Burian, is a masterpiece. It is one of those faces that feels old and new at the same time. The closer you look, the more you discover, but it can still be invisible at text sizes. I have never used it, but I have admired it for some time. The Czech influence might be a fitting connotation for an Orthodox newsletter.

I like FF Scala, but I know many who are older than me that don’t. They tie it to a bygone era, perhaps because it was so popular when it first was released in the ninties. I find it less elegant than the other two. Martin Majoor himself even made a new (version of the) face better suited for literary work: FF Seria.

Maybe I can suggest one of our own as a possible alternative? Masculine and unique. PM me for details. I’ll give you a good offer.

I am considering Maiola, Quadraat and Scala for my needs. I realize that these are postmodern, which is equally foreign to my background.

Dear Father Leonid, I think that your choice of typefaces is indeed… un-Orthodox. Pun unintended. It’s not that those faces are so ‘postmodern’ but they seem to lack that humbleness the subject is calling for.

The main problem with your layout, the way I see it, is not so much the use of flashy fonts as the combination of the breathtaking Trinity with that cross-like typographic кунстштюк. I would keep the type page in the right-hand part of your spread plain, rectangular—maybe same width and same depth as the icon’s image area?

Also, there is no need in inserting the heading between the top two paragraphs. Nor do you have to use quotation marks in a heading. In any event, this kind of quote marks—guillemets/französischen Anführungszeichen/chevrons/ёлочки—is not normally used in English.

Maxim is right the ragged right type block on the right with it's cross like shape takes away from the wonderful Trinity image. Too cute type tricks take away from the overall design.

What's wrong with Alegreya? Maybe slightly more playful than the others you mention, but not all that much more playful than Quadraat. A bit contemporary perhaps, but then again so are your other choices. I like the way it looks in your layout very much. The italic is beautiful and the sans (which I had forgotten existed, actually) works very nicely as a counterpoint. I recently set a lengthy book in Alegreya and was very pleased; its quirkiness and personality worked for the subject … though maybe it's got a bit too much character to use here.

Scala to me doesn't particularly reference anything you're after. It's less flashy but too businesslike. Quadraat and Maiola are both more beautiful, in a craftsy artisanal imperfect sort of way. You need to decide if that's what you want, or if something a little more laid back would be better for the tone of your publication. Have you investigated faces that are more closely linked to the history of Greek Orthodoxy?

The other thing to consider is your layout, which right now is very contemporary. In which case a more contemporary face might be exactly what you need. If you tone down the layout, then you can get away with a more somber face. I'm not quite sure what's going on with the righthand page; I would suggest a more straightforward typographic layout. (I feel like there's a typographic pun here, maybe to do with the cross, but it's sort of going over my head.)

Whether you go with a more traditional or contemporary layout really comes down to what the purpose of the publication is. Maybe you want to be deliberately contemporary because that relates to your mission; I'm not sure. No one says that contemporary design is out. (See this rather infamous edition of the bible: https://www.google.com/search?q=revolve+bible&tbm=isch&tbo=u&source=univ.... Regardless of what you think of the project, there's a designer who understands their concept and goes for it!)