Learn Turkish: Giderek dingin mi dingin bir kartpostal manzarası beliriyor. - Gradually, a very serene postcard scenery is appearing.

This is the exact pattern from this lesson! By repeating an adjective and putting the question particle mi (or mı/mu/mü depending on vowel harmony) in between, you strongly emphasize the adjective. It is much more expressive than simply saying çok dingin (very serene). You can think of it as saying "serene, oh so serene."

Why is bir placed after dingin mi dingin rather than at the beginning of the phrase?

In Turkish, when a noun is modified by an adjective, the indefinite article bir almost always goes between the adjective and the noun (like in güzel bir ev for "a beautiful house"). Here, dingin mi dingin acts as one big adjective phrase modifying the noun compound, so bir naturally slips in right after it.

The word giderek means "gradually" here. Does it literally mean "by going"?

Literally, yes. It is the verb gitmek (to go) plus the adverbial suffix -erek (by doing / while doing). However, when used on its own as a standalone adverb, giderek has taken on the fixed figurative meaning of "gradually," "increasingly," or "step by step."

Why does the word for scenery have a suffix here (manzarası)?