LOKESH DAS

Let F be the Force applied to pull a body of mass m as shown in fig (i) . F Cosθ and F Sinθ are the components of mass F, where θ is the angle between the force F and horizontal. The Normal reaction is given by R₁ = mg - FSinθ . If μ_k is the coefficient of kinetic friction, then friction force is given by, 
f₁ = μ_kR₁ = μ_k(mg - F Sinθ) .......................... 1

If the same force F is applied to push the body the normal...

When a woman in an elevator lets go of her briefcase but it does not fall, then the elevator is accelerating downward with greater acceleration than the acceleration due to gravity 'g' ( i.e. a>g). The reaction, R is negative and the briefcase stick in the celling of lift rather than falling.

Yes, a body can be said to be at rest and in motion at the same time. No body remains absolute rest or motion i.e. they are comparitive body that depends on the reference frame/point. For example: a passenger travelling in a bus is at the state of rest for others passenger in a same bus but, the same passenger is at motion with respect to the pole, tress and houses outside.