There's no problem sending multiple requests to the MultiChain API.
I'm afraid we can't ourselves comment in depth on the Lucid Ocean C# wrapper, but you can always try communicating directly with the node, using multichain-cli on one computer, to connect to multichaind running on another. (If you run multichain-cli with no parameters, you'll see how to pass the relevant options to do this.) This may give you more debugging information, to see the request that is sent and the response that is received, if any.
The most likely explanation is that your node is not accepting incoming API requests from the place you are sending them from, and this can be solved using the rpcallowip runtime parameter for multichaind.