It's just a speculation, but I think régi comes from the adverb rég(en) (a long time ago) it's conjugated similarly as Balaton - balatoni (adjective, originating from Lake Balaton.) This irregularity you noticed is actually the rule in the case of those words, so the plural of balatoni is balatoniak, the plural of szegedi (originating from Szeged) is szegediek.
Mikor volt Ronald elnök?
For "'férfi", the original complete word is "férfiú". It still exists but the shortened version is more common these days.
So, the plural of "férfiú", and of "férfi", is "férfiak". Why not "férfiúk", not sure. Its accusative is still "férfiút" (or "férfi" - "férfit"). But the plural is most commonly "férfiak". Interestingly, the word "fiú" has the plural "fiúk". But it also has a version "fiak", which has a slightly different meaning. Anyway, at least we can see why it is not "férfiek".
Let's see the word "régi". I propose that the "-i" at the end is a suffix.
"Rég" is a word in itself, meaning something like the old times, the past. A version of it is "régen", which I think is also a conjugated form. "Rég-en" - in the past, a long time ago. There are other conjugated forms, which seems to prove that "rég" itself is the root word: "régről", "régtől", "régente", "réges-rég", etc.
So, one conjugated form is "rég-i" - meaning "from the old times", "from a long time ago". That is: "old".
We can contrast this word with the present:
"Most" - now
This one also has a "most-an" form, which is not really widely used these days, except as a carrier of further suffixes, as in "mostani" or "mostantól". "Most-an-i", "most-an-tól". Why not "most-i", or "most-tól"? Maybe it sounded too weird to the ear. Anyway...
The suffix "-i" is a very common one. The genaral role of this suffix is to express a "belongs to" kind of relationship to the word it is attached to:
ma, mai - today, today's (news)
tegnap, tegnapi - yesterday, yesterday's
Budapest, budapesti - Budapest, of/from Budapest
Amerika, amerikai - America, American, or of/from America
konyha, konyhai - kitchen, belonging to the kitchen
So, the word "régi" easily fits this pattern.
The "-i" ending is a suffix.
And, finally, I propose that every time a word ends in this "-i" suffix, the plural will add an extra buffer sound. Specifically, an "a" or an "e". So, the plural suffix will be "-ak"/"-ek", added to the "-i" suffix.
ma-i - will become "ma-i-ak"
tegnap-i - will become "tegnap-i-ak"
budapest-i - will become "budapest-i-ek"
amerika-i - will become "amerika-i-ak"
konyha-i - will become "konyha-i-ak"
rég-i - will become "rég-i-ek".
And if there happens to be a word that does not seem to follow this rule, then the vowel at the end of the word is not a suffix but rather part of the word itself:
bébi - bébik (babies)
süti - sütik (cakes, a shortened version)
(A side note: I have to mention another "-i", which is itself a plural suffix, used in possessive-possessed constructions:
Fiam - My son
Fiaim - My sons
az ő fia - his/her son
az ő fiai - his/her sons
This whole comment does NOT apply to this suffix. )
And an interesting example is
"kocsi" - commonly meaning car these days. It used to be a specific vehicle, from before the motorized times.
Its plural is "kocsik".
kocsi - kocsik.
Why is it interesting? Because it used to mean "Kocs-i", a thing belonging to, or from, the village of "Kocs". This thing was a new type of a horse-drawn carriage invented there in the 15th century, which quickly spread throughout Europe. It used to be called "kocsi szekér" - "cart of Kocs", but then it got shortened to just "kocsi", and became a noun in its own right. So, now, the the plural of "kocsi" is "kocsik":
kocsi - kocsik.
But the people of Kocs are still
This rule with the vowel endings can be extended to other suffixes, as well, but not to all of them. I think it has to do with the type of word and suffix. That deserves another novel. Sorry for being this long, like all the time.
Those are full, intact words in their on right. The "i" at the end is not considered a suffix. Definitely not with "kicsi". And with "nyuszi", let's look at some analogies:
"Nyúl" is to "nyuszi" as "John" is to "Johnny"
The same goes with:
"Sütemény" - "süti"
"Fagylalt" - "fagyi"
"Péter" - "Peti"
"Zsuzsanna" - "Zsuzsi"
"Katalin" - "Kati"
"József" - "Józsi"
"Zsebkendő" - "zsepi"
"Csokoládé" - "csoki"
Grammatically, these are all plain simple words, no suffix there. The reason behind these "i" words is sometimes endearment, other times more like shortening a long word. Btw, there are many other ways to create words of endearment, especially from people's names.