「吾{あ}が舞{ま}へば、麗{くは}し女{め}、酔{ゑ}ひにけり」
舞へば: what is this "へ"? an old verb form ? (I guess the "ば" is the same as in modern Japanese, e.g. "踊れば")
The classical verb is 「舞{ま}ふ」 ("to dance"), so it conjugates to 「舞へば」. Those were actually pronounced 「まふ」 and 「まへば」, respectively.
In modern Japanese, as you know because you are fluent, the verb conjugation is 「舞う」⇒「舞えば」.
This should NOT seem strange because even in modern Japanese, the particle 「へ」 is still used even though its pronunciation has long changed to 「え」. In many languages, pronunciation changes faster than spelling. We still write the word "knight" that way even though we only pronounce half the letters used in it. Every letter in "knight" was pronounced in the old days.
酔ひにけり: is that "酔ひ" ("酔い"?) + "に" + "けり". What does that mean ?
Yes, as we discussed above, 「酔ひ」 later became 「酔い」.
Regarding 「にけり」, 「に」 is the 連用形{れんようけい} (continuative form) of the "completion" subsidiary verb 「ぬ」. 「けり」 is the "past tense" subsidiary verb.
「～～にけり」 in classical Japanese is the equivalent of 「～～てしまった」、「～～たことだ」 in modern Japanese.
My modern Japanese TL of the line in question would be along the line of:
「私{わたし}が舞{ま}うと、（その）美{うつく}しい女{おんな}は酔{よ}ってしまった。」
An English TL would be something like:
"When I danced, the beautiful lady became intoxicated."
Finally..
How old is this Japanese?
The style itself is from as early as 7th and 8th centuries, but it was written in our time.
