You could halo it, or bezel, or whatever you choose. If it's a simple solitaire, perhaps a knife edge shank. https://www.pricescope.com/community/threads/under-1ct-diamonds.25028/ has pics if stones under 1ct. I suggest starting at the most recent and work back, since a lot of the recent e-rings here have been in the .7-.8ct range.
Totally personal preference! I wear 4.25 and love wide bands, or my engagement ring stacked with 2 or 3 other bands. Your best bet is to try rings on of various widths. Best of luck!
I wear a 4.25 and I would go with 1.8 or 2mm - but it's personal preference, nothing wrong with going thinner or thicker if you like that look...I think a relatively thin band makes the stone pop a little more. I would have concerns going too thin, just due to worries about bending or damage. That said, many PS'ers have thin shanks and don't have a problem. I guess I just like a slightly more substantial look. It really just depends on the look you like!
When considering band thickness, I look more at the width of the diamond than the ring size. Since your diamond is 5.78 x 5.82, I think it would look great on a band that's no larger than 2mm.