Why is the phrase 我與父親不相見已二年餘了 correct?
(Let me know if you read traditional characters better! I'll switch upon request.)



  Use of 二年 and not 兩年，I have also seen 二位姑娘，but only in period dramata, so I assumed it was antiquated.



二年馀 is a Classical construction, and sounds formal / literary / etc. in colloquial Mandarin, which would normally say 两年多. But since this is a written text, the author went through solid classical education, and the author was generally not writing in the most vernacular way possible but communicating with some degree of Classical (i.e. written Chinese) tinges, yes there are some Classical constructions that do not appear in modern colloquial Mandarin.



A related phenomenon is that, sometimes people write generally in colloquial Mandarin, but write the characters used in Classical Chinese or formal Mandarin according to the actual meaning, instead of the sound. So you see people write


  我5月12日-6月3日在上海


but hear them say 


  我5月12号到6月3号在上海.


The same thing happens with pairs like 角 and 毛, 元 and 块, 已 and 已经, etc.

As for 二位姑娘, that is a script that actually faithfully transcribes the spoken language, so the issue there is that the use of 二位 instead of 两位 dates to an older era -- which is why you see that in period dramata.







  Use of 不見，not 沒見。Is this the difference between "it's been two years since I saw him" and "haven't seen him in two years"?



I suppose this is readily understood in English actually.


  我與父親不相見已二年餘了。
  
  Me and my father not seeing each other...


There, you just can't mark the aspect of have, or, in Mandarin, 没. 

This syntactic choice (the construction of the whole sentence) is not frequently used, so that is probably why you think 没 should be used there. Indeed people normally say this using this construction:


  我和我父亲已经两年多没见了。




There is also an issue of Classical constructions: 相 + [verb] meaning "to [verb] each other" is normal in Classical Chinese, but no longer active or productive in colloquial Mandarin. People speak colloquial Mandarin, so of course they don't say


  我和父亲不相见...


because you need to do a Classical construction there, and that does not exist in colloquial Mandarin.