Can objects move during a collision-elastic collisions? Can two objects move together during an elastic collision before moving apart? Or would this be considered an inelastic collision since during the time in which the objects were stuck moving together, the kinetic energy was reduced.
 A: Note that terms elastic collision and inelastic collision are just abstractions to indicate certain properties of a collision. To determine if a collision is elastic or inelastic you have to observe kinetic energy just before and just after the contact. If kinetic energy is preserved the collision is said to be elastic, and if kinetic energy is not preserved the collision is said to be inelastic. The only exception to this rule is if the colliding objects merge together to become one body in which case they do not lose contact and the collision is said to be perfectly inelastic.

Can objects move during a collision-elastic collisions?

Yes, objects can move together while in contact during collision. This does not make any difference to determining whether the collision is elastic or inelastic as we are interested in what happens just before bodies make contact and just after the bodies lose contact.
A: Total energy of the system: before and after the elastic collision: $E_b = E_{kb} = E_{ka} = E_a$
During the $\delta t$ of contact: $E = E_k + E_p$, where $E_p$ is the repulsive electrostatic potential energy. $E = E_b = E_a$.
It is not a temporary inelastic collision. What matters is the kinetic energy before and after it.
