Your plants appear to be doing fine. I wouldn't be overly concerned about the color of the roots unless they begin to look slimy as the nutrient will discolor them some. I have grown in hydro for several years and have never used hydrogen peroxide. I did keep it on hand for preventing slime in the water, but with the water changes being fairly regular, I never needed it. It won't hurt to use a small amount but be careful to not over-do it as it can burn the roots if it is too strong.
It will be important for you to maintain your temps in the solution so that it stays very close to 20c to keep the plants happy. I personally would also raise the air temp if possible to around 24-26c as they seem the happiest at that temp.
You will find that the pH will drift over several days as the plants grow and take in nutrients. This is normal and it is (in my opinion) good for them to allow the pH to drift in the range off 5.4-6.3 but don't let it stay at either end of the range for more than a day or two as they aren't able to take in nutrients when the pH is that far out. You will find a rhythm with this where you will set the solution at a point on one end of the pH range and allow it to go for a set number of days before checking and adjusting the pH back to the start again. Early on in veg growth, the pH won't drift as fast because the plants aren't taking in as much solution but once they go into flowering, you will see everything speed up.
I highly recommend that you get a book or one of the large desk calendars and write down everything you do and when so that you can keep up with it. Keep a record of the pH and temps and ppm/ml of nutrients. You will need it later to go back and look at what you did before some problem began or to see how you did something "the last time". I wish you best of luck, and I would like to welcome you to your new addiction because once you start growing your own, especially when you have the first successful harvest, you will not be able to quit