My roux came out a beautiful dark brown. Then I began to slowly add chicken stock. The flour and water did not completely blend and I don't understand why not? The roux was very hot,but the stock and water was room temperature.
There's your problem: slowly. When you add the stock to the roux, an irreversible chemical reaction starts, where the starches from the roux bind to the fluid and gelatinize. After they have gelatinized, they cannot soak up any more fluid. So when you add a small amount of stock and stir it all in, all the starch gets used up, creating a very dense mass. Adding more stock and thinking that it will get absorbed is like pouring juice over jell-o and thinking that you'll get a thinner jell-o.
I know it is counterintuitive - I've done it the wrong way for years and my success rate for bechamel was not much more than 30%. But you just have to get your stock very quickly into the roux, as the reaction is over in maybe 30 seconds. The risk of getting clumps is high, but the sauce isn't completely ruined the way it is when making it too slowly. I even tend to dump the whole stock into the roux in a single pour.
There are quite a few things you can do to reduce the clumps probability, even when going the all-at-once route.
First, make a smaller quantity. If necessary, work in batches. Quickly combining 500 ml milk with the appropriate amount of roux is much easier than doing it for the 1.5 liters you need for the big lasagne dish. Less than 500 ml is even easier.
Second, whisk vigorously all the time while combining. When making a small batch (see above), it is easy to hold a small container with the stock with one hand only and pour from it, while whisking with the other. Pouring some stock with two hands, then picking up a whisk and stirring it in, then pouring some more is a recipe for disaster. It is even better if you don't use a whisk, but a handheld mixer (don't do this in a nonstick pan, even if it is touted as metal utensils safe. If you do it in an enamelled pan, you can get visible lines from the attachments. These are a purely aesthetic problem, but you might want to avoid them nevertheless).
Third, use a high, narrow pan. Your whisking/mixer is only doing some good in the vicinity of the whisk, while the roux in some distance doesn't get mixed with the stock quickly enough.
Fourth (or maybe zeroth, because it is very important), a roux has a very small error margin. Always measure your ingredients exactly, preferably by weight. If using boiling stock, don't let the measured stock boil on while you are preparing other things, as it will evaporate quite a bit.
Fifth, there is the temperature you already mentioned. You want the temperature difference to be as small as possible. Don't set your burner to a high temperature. If you need the hot temperature for a brown roux, err on the side of too cold until shortly before the color is achieved, then reduce the temperature, and wait for the residual heat of the pan to complete the coloring before combining. I prefer to combine on the burner using a boiling liquid, with a temperature difference of around 60°C or less, but have also used the "cold" method successfully: liquid at room temperature or somewhat warmed, but far from boiling, roux is removed from the burner for a minute. Combine away from burner until smooth, then put back on until it bubbles.
Sixth, if you get some clumps, you can save the sauce. If they are few and small (<3 mm in diameter), just stir vigorously until gone (remove the sauce from the burner after it bubbles, then continue stirring). Or use an immersion blender. If this doesn't get rid of them, use a sieve after the sauce is ready (not too fine a sieve, and certainly not a cheesecloth).
Seventh. I find that the lighter the roux, the more easier it is to work with. For bechamel, I often don't brown it at all, just wait until it is hot enough. You may want to perfect your technique with a light roux before starting with the dark ones.
Eighth. Even with a light roux, you will probably fail often the first few times. Roux sauces need exercise until you get them right. So some dry training may be a good idea. Use all-purpose flour, the cheapest oil you can find, and heated water, make some batches of roux and throw them out. The cost will be negligible, and the probability that you ruin the next batch when you are under time pressure and/or are using expensive ingredients (and they can get actually expensive - consider a veloute made with roasted argan oil for 11 Eur/100 ml) is much lower.
You should always add your roux when then stock is boiling. Other wise, you may not only end up in your present situation, but you end up having no idea how thick or thin your soup or sauce will be once it boils. Starches (roux, corn starch) thicken like popcorn pops. Once it gets hot enough, it 'blooms' and causes the thickening of the liquid. If you put it into a liquid that is not hot enough to cause it to bloom, it just sits in suspension (or sinks to the bottom in a lump) until it finally does hit that temp. If you have added too much thickening agent, you will then have to adjust with more liquid, which can adversely affect your soup/sauce.
Hope this helps.
Some people will add hot stock to hot roux without problems. However, if you're having trouble try letting your stock cool before adding it to the hot roux. This will help it combine smoothly (this is what I do). Once you've gradually added the cool stock to the roux mixture you can then heat the sauce. Some people do this the other way around (add hot stock to cold roux).
If you're making gravy, add cold stock to hot roux. If your making gumbo, add not the stock to the roux, but instead add the roux to the hot stock. This makes all the difference. If you add the stock to the roux, you have about 20 seconds and no more water will mix. When you add the roux to the hot stock, you can add slowly stirring at the base of the stock. Every ounce of roux you add is doing it's job perfectly.