Tapering and runout on a SIEG C1 lathe

Hello all,
I have a SIEG C1 Micro Lathe , which I
bought from Axminster a year ago, although the lathe itself doesn't
see heavy use (perhaps, say, once a month?).
Recently, I've discovered 2 problems while turning. The first is that
my work is tapering, with the diameter being larger near the chuck.
I've run a clock along a test piece and have found that there is a
0.15mm deflection across a 55mm length, giving a 0.3mm change in
diameter. The second is that if I place the clock on the work near the
chuck and rotate the chuck, I find I'm getting deflections between
0.03 and 0.15mm, but they don't seem regular - the needle looks like
it's deflecting randomly.
To take these measurements, I put a 10mm diameter, 100mm long mild
steel rod in a 4-jaw chuck with about 60mm protruding and got it
running roughly true. Then I started taking light cuts across the
protruding length (touching on and feeding across). A clock was then
used to take the readings.
The lathe itself is just sat on a stand (seen here
but is not fixed to it at all.
I can't say for sure when these problems started to happen, because
it's only recently that I've started to actively watch out for them. I
have a hunch that the tapering started quite recently when I removed
the lathe's 3-jaw chuck to try out the 4-jaw chuck that I had just
bought. But as I said, I really couldn't say for sure.
As a novice, I'm not sure whether these defects are ones which I can
fix myself or if they're serious problems with the lathe. I was
wondering if I could get some advice regarding these issues?
Thanks
--Amr