Can switching adaptor perform constant current mode?
Q1: What happen when load draw current more than 2A? (Will it preform constant current mode?)


It could go into current limit. It might not and depending on the (lack of quality) could overheat and go on fire. 


  Q2: Is it safe to use wall adapter to charge battery? (Can we consider current on label as charge current?)


No. A charging circuit to suit the battery chemistry and capacity must be used to do this safely. It may be possible to find one that uses your 5 V power-supply as the mains to 5 V DC converter.