Does it necessary to perform handover signalling between two eNodeBs having X2 setup ?

Answer 1

It's optional to perform HO signalling between two eNBs having X2 setep. If eNBs have x2 setup and performing HO using S1 base then X2 path can be used for data forwarding.

Answer 2

It is not necessary that handover is mandatory if two eNodeB's are connected via X2 link. Procedures over the X2 link are categorized into two groups:
1. Mobility Procedures
2. Global Procedures

One eNodeb may support only global procedure , not necessary to support handover as well.
If you go through 3GPP TS 36.423 then these procedures are clearly defined.
Please feel free to ask any further query if you have or expecting more clarity.