No, there is no "N" sound. This is called a nasal vowel. When you have an "N" in a word followed by another consonant, that "N" only serves to nasalize the previous vowel. Hope that's not too technical! :) You can see point number 5 here: http://www.oegugin.com/2014/02/07/silent-letters-in-french/

When letters group together they form new sounds. "in" is one of those group. It does not sound "i" and it doesn't sound "n" either, it's just an "in" sound. By the way, it's miles away from the English "in"! The English "in" sounds like "EENNNN" to me, whereas in comparison, the French "in" is closer to just "è", so very very short! Ps If you read "ainsi" as "aisi", it's not right!

No, it's certainly not a clear consonant 'n' sound, as it would be in the English word 'principle', where your tongue makes contact with the roof of your mouth. In French, it's more of a nasal sound, made in your nose, a bit like the sound 'ahng' .

Pr-in-cipalement, -ain-si, -in-terprétariat, -in-certain, auc-un-, chac-un-, dem-ain-... etc sonnent exactement de la même manière. I suggest you reading more about this difficulty on this site. http://www.jerevise.fr/son-in-mots-phoneme.html In addition, you should also use the Listen Tool on Google Translate for a better understanding of these pronunciations. https://translate.google.fr/#fr/ko/principalement