I glanced at the code and I think the problem lies here
LDA $14 ;\
BNE No_Anim ;|then add 4 to X
LDA $03 ;\
BNE No_Right ;|If facing left,
TXA ;|then add 8 to X
ADC #$08 ;|causing line 3 or 4 in the tile tables
TAX ;|to be used instead of 1 or 2.
You aren't clearing the carry flag before adding which is causing the wrong table value to be indexed. Add a CLC before each ADC.
This should fix the problem.