Will my DIY powerbank be safe? It will be charged using laptop chargers
I want to charge/discharge each cell at 1A max. Is there a chance that
  the the circuit may draw more than this?


Yes, it could draw more than 1A. Your 'USB Quick Charge buck converters' are rated for up to 24W output, so if they were both used to capacity the input power would be more than 48W. 48W/11.1V = 4.3 Amps.


  Temperature switches trigger at 45 celcius. Is that safe for li-ion?


Yes. 


  Is there need for a cooling fan for the buck converter used to charge
  the powerbank? Output will be 12.6V 10A


Assuming the specified 95% efficiency is correct, the converter wastes 5% of the input power. At 12.6V 10A out that would be (12.6*10)/0.95)*.05 = 6.6 Watts. At 10A you would need a cooling fan if there was insufficient natural convection. However since the battery charging current is limited to 1A the converter will never have to deliver more than 1A, so it probably won't get hot enough to need a cooling fan. 

Overall your circuit looks like it will work, but is it safe? The only issue I can see (apart from making sure the charging voltage and current are set correctly) is that you have no over-discharge protection apart from the BMS. If the BMS low voltage cutoff is below 3.0V per cell then the battery could get discharged to the point where it should be recharged at a lower current.