Why is “mycorrhiza” translitered with two “r”s?

Because had the word existed in Ancient Greek, it would have been written with two r’s.

Ancient Greek had a rule that if anything was prefixed to a word starting with r-, the r- was doubled. That did not involve just compounds, but also prepositions put in front of verbs, alpha privative (the equivalent of un-), and even augments, the prefix indicating past tense:

  • μέλι “honey” + ῥυτός “flowing” > μελίρρυτος “flowing with honey”
  • ῥέω “to flow” > ἁνα-ρρέω “to flow back”
  • ῥευστός “in flux” > ἄ-ρρευστος “static”
  • ῥύω ” I cleanse” > ἔ-ρρυα “I cleansed”

The rule had broken down in spelling by Byzantine Greek, and the rule has no application in either pronunciation or spelling in Standard Modern Greek. But of course scientific use of Greek took Classical Greek as its model.

So when new compounds were made up in scientific Greek in the West, they (usually) applied the rules of Classical Greek phonology to their coinages.  After all, Classical Greek had a -rrhiza compound already: γλυκύ-ρριζα “sweet-root” = “liquorice”. Liquorice, scientific name: Glycyrrhiza glabra.

What other languages influenced Greek?

In terms of the usual interpretation of the question (what languages did Greek borrow words from), at different times Greek has borrowed words from:

  • Persian (a small number)
  • Latin (a fair few)
  • Slavonic (surprisingly few)
  • Albanian (surprisingly fewer)
  • Aromanian (ditto)
  • Catalan (one word, παρέα < pare(j)a)
  • Romany (very few, although it is the go-to source for cants (secrecy languages), including Kaliarda, the gay cant: Roz Mov – Kaliarda
  • Old French (Cypriot)
  • Italian and Venetian (lots, though they have been purged)
  • (Ottoman) Turkish (lots, though they have been substantially purged)
  • French (before World War II)
  • English (after World War II)
  • … oh, and Ancient Greek
  • Can’t think of any words from Aromanian, but there should be a couple

Dialects outside of Greece have borrowed from the majority languages: Calabrian and Salentino in Southern Italy, French in Corsica, Russian and Tatar in Eastern Ukraine (formerly Crimea). And of course languages in the diaspora do the same.

How different is the Ancient Greek language from the modern Greek language? Can any Greek-speaking people testify if they understand classical Greek of Homer, et al?

I understand most of what’s going on in the Gospels, though much more so with Mark and John than Luke and Paul. Some Attic texts (and the Byzantine texts emulating them) are a challenge, not least because of their abstruse syntax, but I still have  a hazy notion of what’s going on. The syntax in Homer is much easier, but the vocabulary is impenetrable to me.

I’m an odd special case: although I am a self-styled world expert on machine recognition of Greek, I have not studied Classics, so my vocabulary is behind those who have studied it in school or uni. (In fact, for that reason my Aeolic is probably better than my Homeric — I had to do work at the Thesaurus Linguae Graecae on recognising Sappho, while Homer was already taken care of.) So my Ancient vocabulary starts from a more naive base from some other answerers’.

Greek has been more conservative than other languages in Europe, and part of that was the influence of Ancient Greek via the Church. (In many parts of Cyprus, for example, /θ/ has changed  to /x/ throughout the language; so anthropos is pronounced akhropos. With one exception: the word for God, theos.) Universal literacy in Iceland had an even starker conservative effect. The grammar has also been conservative, though there is not as much grammar as there used to be. (No dual, no infinitive, no optative, no perfect, no future, no dative, no third declension, just two conjugations.)

Bear in mind though that there was massive reimportation of Ancient vocabulary and phrases into formal Greek (in no small part to eliminate Italian and Turkish loanwords); and that Greek has an historic orthography. So Greeks now can read more Ancient Greek than they were actually supposed to; and an 18th century Greek peasant, time travelling to Ancient Greece, would have little idea what was going on.

How different are Cypriot names from their Turkish and Greek counterparts?

Greek Cypriot surnames are often patronymics, formed as the genitives of given names. Surnames are quite region-specific in Greek, so you can tell a Greek Cypriot surname: it’s the one *without* a suffix, like -opoulos, -akis, -idis, -ellis, -atos, etc.

Greek Cypriots use a few more Ancient names than Greece Greeks, and a lot more Old Testament names. For a truly random sampling, there’s the current Cypriot cabinet:

           

  • Nicos Anastasiades
  • Ioannis Kasoulidis
  • Harris Georgiades
  • Socratis Hasikos
  • Christoforos Fokaides
  • Costas Kadis
  • Marios Demetriades
  • Georgios Lakkotrypis
  • Nicos   Kouyialis
  • Zeta Emilianidou
  • Ionas Nicolaou
  • George Pamboridis
  • Nikos Christodoulides
  • Constantinos Petrides

Nicolaou is “Nicholas”; that’s in fact my surname in Greek. (My father is Cypriot, though I haven’t spent much time there.) Most of the other surnames are -ides/-ades, the revived ancient Greek patronymic which also got taken up by Pontic Greeks. Ionas is “Jonah”; you’ll be hard-pressed to find anyone in Greece called Jonah. There are some Socrates’s in Greece, but I think there are rather more in Cyprus. Btw, Marios (Mario) is more common in Cyprus as a name as well.