I wouldn't mix tiger barbs with the harlies and cories. They are really active and greedy at feeding time and the harlies may not be able to get food. If you want tiger barbs, get 8 of them and then get 6-8 of a semi-aggressive tetra like Serpaes. The black widows or Emporer tetras would work instead of the barbs.
I've heard of tiger barbs harrassing cories...so I wouldn't try it. How long is your tank? You may be able to have some loaches instead of cories. They would be a better tankmate for barbs, since they aren't quite as peaceful as cories. There are some smaller types like Dwarf Chain loaches, Yoyo loaches (if your tank is long), Polka dot loaches, and Zebra loaches. They are really active though, so if your tank is less than 3 ft, I'd stick with the cories and change out the barbs for the tetras.
8 cories will be fine. They only get 2 inches (pandas or bandits), so 8 would be ok in a 45g tank.
I believe rams will eat the shrimp, so I'd leave them out, unless you just want to buy some ram food...lol.
An apple snail may get picked on. I've never kept one, but since they get bigger than my ramshorn snails (which the ram leaves alone), they may get bullied. I'd go with MTS if you plan to have sand or a fine substrate. They bury, so are great at sifting the sand and help to avoid anaerobic pockets. They are also harder for fish to eat.
I know some people keep white clouds in with other tropicals. Just keep a lower temp., like around 74F. I've never kept them, so I don't know what temp. they like. You could substitute emporer tetras. I think they are really pretty. You will probably have alot of fry...the probably won't all survive with a ram in the tank, but you could get lucky and have some survive, which will increase your stocking. I'd go ahead and have 4-5 Emporers instead of the white clouds.