Your roux was slightly overcooked -- for the pupose. It separated because the flour became so toasted it couldn't even hold oil.
You can make a perfectly good roux with oil, it doesn't have to be butter. The "best" choice depends on what you're doing with the roux. By and large I'd say butter is the better choice. Or is that better is the butter choice? I'm easily confused. But when it comes to gumbo, etouffes and other cajun stews -- oil is the most common choice.
Of course it all depends on what you're trying to do, but "peanut butter" (a light to medium brown) is probably about as far as you want to take your roux. That is, at least until you have a good handle on making gumbos.
Darker rouxs have different, deeper tastes; but as resident1fan said, darker rouxs don't thicken as well as lighter ones. It's better to start with something that responds well to a more basic skill level.
FWIW, the technical term for a light roux is "blonde," and not "white." Not that "white" is wrong -- it's a regional (and technical) thing. "Peanut butter" is also a technical term -- at least in Cajun cuisine.
You can combine roux and liquid in the following ways:
- Hot roux into hot (simmering) liquid;
- Hot liquid over hot roux;
- Cool liquid (initially added in small stages -- whisked until it heats and thickens) over hot roux; and
- Cool roux into hot liquid.
What you cannot do is combine cool roux with cool liquid or you'll get lumps. Warm roux combined with warm liquid isn't much better.
When adding stock to roux (which is usually how gumbo is made), even using hot stock, it's best to add a little at a time and whisk each addition until it's fully thickened and smooth. Most cooks use a high flame, and moderate the temperature of the mix by adding the simmering, warm or cool liquid. After you've got your gumbo just a tad thinner than your desired consistency you can reduce the heat to a bare simmer and let 'er cook.
Generally speaking, roux thickens best at the boil. If you add hot or cool roux to a pot of liqud. A short period of boiling and mixing -- two or three minutes -- is a good idea with a blonde, peanut butter, or medium brown roux. However a boil will break a darker roux -- which is quite likely what happened to you.
Hope this helps,