You bring up some interesting points. I think a difference of 50 K in temperature seems a bit high & one could do things to reduce this difference- simply painting it white would reduce temperature difference. But this would a problem that would need to be solved because even small differences would be serious problem. One might even do strange things like beaming microwaves to control it's shape. And perhaps this system could used to counter this 5% libration problem (which sounds like a more fundemental problem which at the moment I can't think of another solution).
And maybe these types problems would have a major influence on where a tower was build. For instance, for the tower and with the much longer elevator cable, the farside might be a good location. Or this tower might be mostly about putting up solar panels in the polar region and for radio reception and broadcast to the polar region.
As far as making a tube shape; it might wiser to use standard pipe sizes- use thousands of miles say 2 diameter pipe, but I've been thinking how to make a more customized cone shaped pipe.