Equivalent circuit of a transformer
Are Lp and Ls only due to leakage flux or they generally represent
  self inductances of both the coils? If yes, what about the rest of the
  flux, won't that induce emf?


Lp and Ls are leakage inductances due to imperfect coupling between the primary and secondary. Lm is the prime mover when it comes to flux because it is the magnetization inductance: -



Above image from here.


  How can the induced emf leads the flux by 90∘


The current in Lm is 90∘ lagging the primary voltage because 

$$V_{primary} = L\dfrac{di}{dt}$$

Current and magnetic flux (\$\Phi\$) are in phase but, there is a further 90∘ shift in the induced secondary voltage due to: -

$$V_{secondary} = N\dfrac{d\Phi}{dt}$$


  Why does resistor reduce the lag in a RL circuit?


For a very high frequency stimulus the inductor would be seen as open circuit compared to R hence, as frequency rises, the 90∘ lag gets smaller.