To answer the last one, yes.
On some types the reverser plume interferes with the aerodynamics, either of the wing or the tail (depending on engine location). If the aircraft has limiting or marginal xwind capability, this plume interference may adversely affect the handling, further restricting the capability.
On Challenger 604, for example, the crosswind capability without using TRs is 24 knots DEMONSTRATED and no explicit Limitation exists. With use of TRs that 24 knots becomes a LIMITATION.
On other aircraft, limits are specified on the level of reverse thrust which may be used, in part to ensure adequate controllability is maintained. Loss of directional stability in a crosswind would be unpleasant. (Note - I doubt there is no stability at 24.5 knots - but the OEM and the cert authorities doesn't want you to go trying it to find out!)