I was just wondering what the essential differences are between these and how I would use them in everyday speech. Thank you.
You've got 3 words here that look similar but in meaning and use are really pretty distinct.
どうか is used to make very polite requests. It has kind of a strong meaning so I don't think it's something that comes up too often in your day to day life. It can have a kind of sense of desperation to it, like begging a doctor to save someone's life or begging someone not to take you to court or something like that, or it can also just be affixed to polite requests. (どうか助けてください！） (alc for reference)
どうにか expresses the idea of "somehow or other," like you're not really sure how or why it worked, or you didn't think it would. Basically things just weren't that great but some way or another you managed to pull through. (難しかったが、どうにか成功しました) (alc for reference)
どうかして kind of borrows from どうか but only insofar as it describes doing "something." どうかして just means that you want to do something, somehow. In fact it doesn't really matter how, you just want to make it happen. It may be more useful to think of it as a hypothetical sort of situation like in the example below, like "if by chance something were to happen." Examples on alc seem to show this having a kind of poetic "if only" kind of feel to it, like this:
If only to-morrow on awaking, I could again call all to mind so vividly!
This is not to be confused with どうかしている, which is an expression that means you've gone crazy or that there's something wrong with something/someone in the not thinking straight sort of sense.