Aww, sunnydawnie, I'm so sorry you lost Jewel!
I know how much you loved her and how proud of her you were. Your friend with the fish knowledge is right, it sounds like Jewel had an internal parasite. Red poo doesn't necessarily mean a problem - my bettas sometimes have red poo after eating a bloodworm. But white poo is always a warning flag.
I don't know how long it takes parasites to show up so I can't say if Emerald is likely to be infected or not. Take him out of his tank and do a complete water change, rinsing the sides and decorations with very warm to hot water. Go ahead and get some parasite tabs and keep on hand but I would wait to see if he develops any signs first; someone with more experience may have better advice. If he starts to lose his appetite, become lethargic, have white poo, or bloat, then start treating him. Remove any carbon from your filter first.
When your tetras and barbs arrive, you might want to quarantine them for at least a week to make sure they are healthy and are not bringing any parasites in with them. They will be stressed from their journey so their susceptibility to illness will be greater. Also, that gives you time to see if Emerald is sick and to treat him.
I personally never use anything but warm water and vinegar to sterilize a tank or bowl. The risk of any bleach being leftover is too great and it could kill or harm a new fish. If you're absolutely certain you got all the bleach out then that's great.
What kind of barbs are they? Tiger barbs need to be in large groups or they will be relentless nippers. And by large, usually 8 or more. You might get away with 6 but the more the better for tiger barbs.