Is momentum conserved when 2 carts having the same mass & the same speed collide, stick together and stop?

For part A. The momentum is always conserved. Since momentum is equal to mass times velocity, one of your carts had a negative velocity and one had a positive velocity. When you add them up for the total momentum at the collision it is zero because they cancel each other out.

For part B. The momentum of the system is the same. Each cart just translated their momentums to each other so the velocities flipped. Good luck!

good and important Q.

Based on F = mA which is really F = dP/dt, with P = mV

AND on action Reaction Fa = - Fb exactly

the sum of Pa and Pb before = exactly P after. after va= vb since they are stuck

in elastic va =/= vb

in elastic kenetic Etotal before = E total after but both P and E a,b have exchanged

in in elasitic some Kenetic E has changed to heat in the "sticky" collision

P is always conserved, Kenetic E may be conserved and Total E is conserved, (including heat)

In theory it is simple but in actual practice a lot of minor effects make it hard to predict