i have 5 tigers in a 20long with a brig and 3 nerites and they dont pester the snails. When i had some ghost shrimp i noticed that they didnt last very long, so i assume the barbs were pestering them. I didnt have many hiding spots at the time, and now that i have a much heavier planted tank i plan on trying a few shrimp again. And it looks like those are the green/mossy tiger barbs you have.
i don't think cories would be a problem but i'm not sure. you could get away with snails possibly but i'm not too sure of them either. i had my barbs for a couple of months and after adding/researching my options just seemed to get shorter and i wound up taking mine back. the lfs i took them to had them in every other tank already. that made me nervous so i went back the next day thinking i was going to have to buy them back at twice what i sold them for to save their lives. i was relieved when one of the employees noticed me right away and just had to tell me one of their long term regulars who has a house full of tanks saw them and was astonished how healthy they were and just had to have them. apparently he bought the frog fish they had there also.
long story short i really have no idea what they are compatible with but i can make you a list of what isn't going to work out so well... basically everything in my tanks except my sae's and maybe my bristlenoses and snails.
I personally think so. But, I stock my tanks on the heavier side. If you go with that stock list, weekly water changes of at least 30% are an absolute must. You will not be able to slide on that. You also will need to make sure to feed the corys a sinking pellet or wafer occassionally. The tiger barbs WILL eat everything you throw in there. Not much will reach the bottom.
I know you don't want to take the barbs back. I love mine too. They are very entertaining. However, from someone who really likes them...I have 4 in a 30 gallon and they really limit what your other stock can be. Just a thought.