I have a flat obijime made of two cords woven together, and it's white and silver lengthwise; is it appropriate to use with a yukata? Also, I saw an infographic saying that tucking the ends of the obijime up is for celebatory occasions, down is for mourning, and one up one down is informal; is this correct? Lastly, my yukata fits me pretty precisely and the tucked part tends to rise underneath the obi so it isn't visible; is this okay?
Usually, yukata obi do not require obijime. Most yukata are paired with hanhaba obi (narrow belts), tied with knots (like this one) not calling for obijime use:
Obijime use is still possible though! But to keep it seasonnally nice, the best would be to use a summer obijime, which are kind of mesh and airy looking:
By the description you make of your obijime, I’d say yours is a pretty formal one (white and silver are formal colors) and not the best to use with informal yukata. The infographic you saw is indeed correct:
Woman kimono have to be dressed with a ohashori, a tucked part at the waist. But it’s only needed so the kimono sits nicely on wearer’s body and doesn’t unravel during the day. It’s better if it shows but not mandatory :)