"Is it necessary to do Stress Analysis using CAESAR II?"
This is wholly dependent on the discretion of the client/end user, or an engineer / stress analyst experienced in these matters. With that said, I agree that yes these temperatures would require an analysis.
You will need to verify that the nozzle connections (that you modeled as anchors) do not exceed the allowable forces for the equipment they are attached to. If the equipment is new, you typically can get the manufacturer to verify your reactions. If they aren't new, well, you'll have to dig for this information.
(Your pipe might pass code checks, but it's a bad design if it rips your nozzles off your vessel.)
As far as the tee is concerned, you will have to do *something* to simulate what it does to your system. Does it expand and contract like your pipe does? If so, you'll at least have to add its deflection to your piping.
Finally, you may wish to consider a flange check on the nozzle depicted on the left.