Dagesh question

Miriam at The Floating Bear asks whether the proper pronunciation of a term is "Darchei Shalom" or "Darkei Shalom"

I'm not a dikduk nerd, but I occasionally pretend to be one, and I've wondered about similar questions, myself, so decided to look into the issue.

Usually, a letter is degusha when it appears at the start of a word or the start of a syllable following a closed syllable. This is one of the exceptions, though.

When most masculine plural words are nism'chim, a sh'va is put under the penultimate letter of the root and the last letter of the root is given a tzeireh. Hence, d'rachim would become d'r'chei. Due to the awkwardness of two consecutive sh'va'in, the first sh'va is changed into a tenuah ketana and the second sh'va is known as a sh'va merachefes, which is unsounded like a sh'va nach, but does not make the letter immediately following degusha, like a shva na does. Hence, the proper pronunciation would, indeed, be darchei (as the word appears in Mishlei 3:17).

There are probably some errors in this analysis, but I think it covers the gist of the issue accurately.