Etymology and pronunciation of arch-, archi-

Solution 1:

An Etymological Dictionary of the English Language (1888) by Walter W. Skeat answers all your questions at once.

ARCH-, chief; almost solely used as a prefix. (L.--Gk.) Shak. has 'my worthy arch and patron,' Lear, ii. I.61; but the word is harshly used, and better kept as a mere prefix. In arch-bishop, we have a word in very early use; A.S. erce-bisceop, arce-bisceop (Bosworth). Thus arch- is to be rightly regarded as descended from A.S. arce-, which was borrowed from Lat. archi- (in archi-episcopus), and this again from Gk. άρχι in άρχιεπίσκοπος, an archbishop. Gk. άρχός to be first; cf. Gk. άρχή, beginning. Cf. Skt. arh, to be worthy; Curtius, i.233. The form of the prefix being once fixed, it was used for other words. Der. arch-bishop, arch-deacon, arch-duke, arch-duchy, &c. In the word arch-angel, the prefix is taken directly from the Greek; see Archi-.
ARCHI-, chief; used as a prefix. (L.--Gk.) The older form is arch-, which (as explained under Arch-) was a modification of A.S. arce-, from Lat. archi-. The form archi- is of later use, but borrowed from the Lat. directly. Gk. άρχί-, prefix. See Arch-. Der. archi-episcopal, archi-episcopy, archi-diaconal. In the word arch-angel, the final i of the prefix is dropped before the vowel following. In the word arche-type, the prefix takes the form arche-.

Another edition of this dictionary (revised in 1910; reprinted in 2005) adds a couple of comments to the explanation.

ARCH- <...> In the word arch-angel, the ch remained hard (as k) in the Romance languages, on account of the a following; cf. Ital. archangelo, Span. arcangel.
ARCHI- <...> The same prefix also forms part of the words archi-mandrite, archi-pelago, archi-tect, archi-trave.