I don't know about their legitimacy, but either way it seems clear that mining is unlikely to be profitable for those without special access to cutting-edge mining technology. I'd recommend attempting to get a refund if you can; you won't make enough mining to justify the risk of a delayed or cancelled shipment.

Unless you have special knowledge or insight, you will not be able to accurately predict the relative prices of Litecoin and Bitcoin. If there was a well-known formula for predicting that Litecoin will soon be worth .04 BTC, then the price would already be at .04 BTC as everyone followed the formula. Trying to predict the relationship between two EXTREMELY volatile assets is a very bad idea. Trying to time this market is simply gambling and I would strongly urge you not to start down this path, ESPECIALLY with money you cannot afford to lose.