Handwritten shape of 「舟」 when it appears like 「月」 and the component form of 「肉」
(Part 1) Was the component form of 「舟」 ever differentianted from 「月」 in handwriting (from 楷書 onwards)?
  


If you count handwritten dictionaries or volumes, then yes, they're usually recorded as variant characters. See a 國學大師 record:




  
  (Part 2) It seems that there were efforts to differentiate the various origins of 「月」 in a character, but were those efforts ever widespread?
  


If you're talking about handwriting, I don't believe that this is true in a mass-scale sense:



Excerpt from a Song Dynasty version of Shuowen

If you're talking about the print styles in the Kangxi dictionary, I'll draw your attention to the layout first:



As you can probably infer, Kangxi print styles made a conscious effort to incorporate Shuowen small seal script features. General print shapes before or even after Kangxi did not really do this.



Excerpt of 《論語主疏》 printed in the fourth year of Qianlong, from ctext


  Would it be seen as strange for someone make that distinction in their handwriting today (although the usefulness of actually doing this can be debated)?


We can take the ROC standard as an example, because we know that they differentiate between「月」and「⺼・肉」...



...but not with「舟」!



However, the effort in distinguishing them results in such a small change in the print shape that, if you carry this on in handwriting, I doubt that it'll be seen as weird. It'll be kind of weird if you fully wrote「舟」though, I'd imagine that someone not familiar with paleography might not even recognise the character.