The not going fullscreen is not necessarily that easy to fix (and it would go black around the window and have it centered for fullscreen usually, not sure why it is showing desktop), but the going top right thing might be related to a persistent bug that happens with p. much all sonohanas and isn't necessarily easy to avoid. What you can try for that is changing compatibility modes to xp or something, and maybe running it as administrator.
Making it go actual fullscreen instead of going original size is possible, but you'll need to change some settings in your graphics drivers most likely; you need to change so that it preserves aspect ratio (or stretches, if that's your jam) when fullscreening things. I know this is possible for Intel GPUs as I've done it on my laptop.
My current sorta-working setup on my laptop involves using compatibility for windows xp service pack 2. That said, it randomly crashed when I actually tested it after like 5 minutes, so uhh... just know that in general the sono hanabira series has a few bugs.