How was each letter of the alphabet called in Latin?

Merging various sources and adding an abnormal quantity of imagination, I
came up with this tentative list:

A [a]
B [be]
C [ke]
D [de]
E [e]
F [ef] (?)
G [ge]
H ['hakka] (???)
I [i]
K ['kappa] (??)
L [el]
M [em]
N [en]
O [o]
P [pe]
Q [ku]
R [er]
S [es]
T [te]
V [wu] (?)
Y ['ypsilon] (?)
Z ['dze:ta] (?)

But I have lots of doubts, especially for those names which I marked with
question marks.

My biggest doubts are with letters "H" (Was it really named ['hakka]? If it
wasn't, where do Italian "acca", and French "hache" come from?) and "K" (Did
it really have a Greek name? If it did, what's the reason? Unlike "Y" and
"Z", "K" has always been in the Latin alphabet?).

Moreover, what was the gender of letter names? which declensions did they
have? were the vowel of monosyllabic names short or long?

And why do some letter names ("L", "M", "N", "R", "S" and, most puzzling,
"F") have vowel [e] before the consonant rather than after it?

Finally does "modern Latin" have names for modern letters derived from "I"
and "V"?:

i
j
u
v
w

Thank in advance!

_ Marco