1. Would a gourami be ok with the other fish I have listed?
I'd agree on the 'it depends on the species of rasbora' point. Things like harlequins do great with three-spots but it;s best to avoid very small species like spotted rasboras. The rest I think would be fine. With danios it can be tricky because they occupy the upper layers so compete with these gouramies for space. You may want to reconsider them for this reason. They can usually avoid the gourami though.
2. Would having only one gourami be ok? If so, should I get a male or female?
I second the male suggestion. Because females are social, they do better in big tanks in large female-only groups. Males are territorial so happily live alone.
And, yeah - no pairs. Three-spots are one of the most aggressive of the 'common' gourami species and males will chase females continuously. They can be kept in large groups with several females per male but then you would have an overcrowding problem. Also, if these fish spawn, the males get extremely aggressive and will even kill non-gourami tankmates - including things like cories that most other fish would ignore. Two males is an absolute no-no as they'll usually fight to the death and two females is problematic because the more cominant will chase the other and this will stress her. the result is usually disease, followed by death.
3. How fast do blue gouramis grow?
Very fast compared to most fish

That's vague lol but this depends on how much you are feeding, what the food is and what the temperature of the tank is at. High temperatures, rich live foods and several feedings speed up the rate of growth. It is also important that the fish is not stressed and that it is disease-free. The frequency of water changes is also crucial to maintaining a healthy growth rate and prevention of stunting. Male three-spots grow to 6", females to about 5".
About shrimp - none of the non-gourami fish you are considering will touch the majority of shrimp species (cories included - mine actually sleep in the same cave as the shrimp

- very cute). I have also kept both cherry and amano shrimp with three-spots without problems. However, young, small shrimp may well get eaten and they do best in a heavily planted tank. With amanos, that just means you need to buy larger ones. With cherries, it also means you won't be able to breed them dispite the fact that their offspring can be raised in freshwater (amanos require brackish). Because they shed their 'skin' as they grow, shrimp are also vulnerable at specific times so need to have plenty of hiding places to avoid being harassed or eaten.