What's the difference between frisk and fersk?
'frisk' can refer to a person being healthy or fresh fruits and vegetables. 'fersk' would be used for newly harvested products, but mostly meaty ones like 'fish'.
Friskt i hukommelsen? No?
No "mitt" in group of words for translation. I guess in this case norsk does not need to use possessive pronoun for "my memory".
Correct. "Å ha X friskt i minne" is a set phrase, so adding a possessive sounds a bit off.
It's very natural to omit the possessive in English, too:
"I have the [experience] fresh in mind."
Why is it minne and nit minnen?
it's an old dative expression, just like 'i tide' (in time) and 'i søvne' (in sleep).
(Though, unlike 'tid' and 'søvn', 'minne' ends with an 'e'.)