I am currently doing a flow problem which involves convection in a rect. enclosure. My vertical walls temperature are at a non-dimensionalised 1 and 0 on each end respectively. temperature at top is 0.5 while the bottom wall is insulated. However, my temperature plots shows that there are certain points which are negative in value. Is this possible?

Physicaly, this is impossible if you have no source term. But, it is possible to obtain such values with numerical methods. This might be caused by improper discretisation of the convection term. In order to ensure stability, numerical dissipation must be added. It can be explicit or implicit. If precision is not an issue, I suggest using an upwind sheme. If you want more information, you could give us the type of method that you use (FDM, FVM, FEM, FVEM, ...).

For the vorticity transport equation, ADI scheme is ultilised. GS-SOR was used for the streamfunction. As for the energy transport equation, i am using point iterations. Could it be that i have to use ADI for the energy transport equation as well to get unconditional stability?

No I don't think so. Since the space discretisation nmight be the cause of you problem, An upwind scheme will give good results..