Hi "BaLLer9". I'm going to try to bring together what others have said as it may sound like we're all disagreeing but really there are a few key messages...
1. You really should ensure you have a full assessment by a physiotherapist. They will be able to tell you the state of the ligaments in your ankle by doing appropriate stress tests. (i.e. are they lax / torn / relatively normal) These you can't do yourself.
Based on this, the therapist would be able to give you a better idea of your prognosis...(how long a recovery is expected / is bracing necessary / is a surgical consult necessary).
They will also be able to appropriately progress a rehab program for you.
2. If you are willing to do your own rehab program at home THAT'S GREAT! This type of exercise doesn't always need daily supervision from a physiotherapist. It just requires that you can have follow-up for reassessment and progression. In between appointments, you can continue to work your butt off at home!! And yes, daily commitment to exercises is important (3 days a week doesn't cut it!).
3. Although icing, water, strapping may make it feel better particularly after a recurrent sprain, it won't "make it better". They are all good tools to get you comfortable through your day but it is still going to take exercise on your part.
4. The rehab program you made for yourself is a good start. But they are very much low level exercises and gradually should be progressed to get to more sport-specific movements required for basketball. Again....it would be very useful to consult a physio (in person) for some guidance after an appropriate assessment.
5. As for your concern about surgery.... There is absolutely no way anyone can tell you if you need surgery or not unless they fully assess your ankle (which again, needs to be done in person). If it's any consolation, I have not seen many ankle sprains that have needed surgery so it's definitely not the norm. However, you REALLY need to have this ankle assessed, listen to what the findings are and ask all the same questions of the physiotherapist that is assessing your ankle.
Good luck!