I am sorry for the delay. It soulds like you may have a fungal infection secondary to the irritation. I would recommend you mix some Lotrimin cream ( antifungal) with some cortisone( helps with swelling and pain) and apply to the area twice per day, cover with cotton gauze to prevent the toes from rubbing together. If this doesn't improve in a day or so , see your Doctor. Infections of the feet can become serious. I hope this helps, please let me know if I can help further:-)
All the things you want to too are too caustic for the condition you are describing.( especially tea tree) It sounds like fungus because fungus splits and causes bleeding. It's the most likely diagnosis because by picking at the area over and over all of the normal flora was destroyed and fungus ( which is abundant everywhere) culod easily invade the area. I would really urge you to just use the two creams , atleast today and tomorrow. Youi will almost certainly see improvement.
Signs of an infection will include pus, redness, swelling.