Wow! A late flurry! Thanks for theses suggestions...
Yes - I play better sitting down, especially finger style. It's partly down to being better connected to the guitar, but mostly, I suspect, dueto the left hand position being better. The shoulder thing isn't because I move my shoulder - I just seem to hunch it a little. But I have shortened the strap to attain (nearly) the height it sould be if I were sitting, and - spooky this - sticking with it and practicing and trying to keep relaxed seems to be doing the trick.
As to the strumming thing - I have played for, um, over three decades, and have always championed in my small way the "art" of strumming. Just like fingerstyle, there's good and bad strumming: rhythmic/arhythmic; exciting/boring; etc. Just because strumming is where many start out it gives strumming a bad name. I like to vary my sets to include some of each of the few different styles in my repertoire which are presentable.
And the tuning when the strap is attached to the nut issue - I've had guitar's like that! Happily my current two (including a twenty year old workhorse of a Yamaha FG-something-or-other) have a neck which is pretty stable.
Thanks again for late flurry and positive suggestions. An if you're near Reading, Berkshire, England on Sunday, come and hear how I'm getting on. Headline gig at Readifolk. More gigs at my site