(C&C note/note to David: I'm not sure you should attempt to use
'hajimemashite' as a question. That's more of a greeting, IIRC.)

The literal meaning is "it's a beginning", meaning that this is the
first time these two have met.  The first meeting is the ONLY
time either party should use this word.  (When I was first learning,
I thought this was a general-purpose greeting, and used it on a
gentleman I'd met before, to his confusion.)

English language...  Eigo.  [Ay goe]"

Pronunciation disagreement here. 'ei' has more of an 'ey' or 
'eigh' sound,
as in 'hey' or 'weigh', than an 'ay' sound, as in 'say' or 'pay'.

(blink)  There's a difference?  Not the way I speak...

When
speaking Japanese, you should take care to speak each 
syllable with the
same duration, 

Yes.  This is VERY important for vowels.  Lengthening a
vowel sound usually changes one word into another.

"Just so you know, the word for yes is 'hai [high (as in tall)]'"

"Hai," I replied.

(And as is frequently noted, 'hai' is NOT a general-purpose 
'yes'. It's
both formal, and has numerous other uses, which usually 
involve responding
to a sensei, or to a superior. When familiarity sets in, 
unless you have a
peculiarity for mixing formal and informal speech--and this 
DOES happen to
people; I know from personal experience--your best bet is to remind
yourself to say 'ee' instead of 'hai'.)

And, as is often noted, 'hai' is often better translated 'I got that',
or 'I understand'.  The combination 'hai, chigau' actually means 'no',
since it literally means something like 'I understand (but) it's
different.'