How to pipette accurate volumes from BSA solution without air bubbles?
Never vortex the BSA solution. The best way to prepare a BSA solution is to put the BSA powder on the buffer solution (or water) instead of the other way round. Tap or rock the tube to dissolve the BSA. If there are some tiny clumps then pipette very gently (use 1ml tips; cut the tips if needed) but tapping/rocking should usually do the job. 

From CSHL protocols:


  For a 10% (100 mg/mL) stock solution of BSA, dissolve 1 g powdered
  Fraction V or molecular biology grade BSA in 10 mL of distilled H2O;
  to avoid clumping, dissolve by layering the powder on the surface of
  the liquid. Gently rock the capped tube until the BSA has dissolved
  completely. Do not stir. Store in aliquots at −20°C.