Edit: The question has been updated and the plot showing a "blip" in inclination which has the same shape as the "blib" in eccentricity. I've incorporated my comments here:
The only phenomenon I'm really proposing is that when limited measure data is fitted and parameters are extracted, there are sources of noise in the extracted parameters, and the noise can be enhanced when there are statistical correlations between parameters. Put simply, a simultaneous change in two (or more) parameters in a correlated way can result in a nearly identical goodness of fit, and this can amplify the ever-present noise in the initial measurements.
You've now flagged two parameters that changed at the same time (eccentricity and inclination on 28-01-2018) and in a beautifully correlated way. The "blip" in inclination matches the shape of the blip in eccentricity. This is a textbook example of correlation in fitted parameters! Applying Occam's razor to the choice between a well-understood mathematical behavior when fitting imperfect, under-sampled, noisy data, and conspiracy theories of "secret spacecraft reactivations", and what's going to be the most likely explanation?
The way to proceed in a systematic and scientific way would be for you to look at all parameters (rather than "revealing them" one at a time), plotted over much longer periods of time, for many spacecraft in similar decaying orbits. I am guessing you will find blips and wiggles are the rule, rather than the exception. Public TLEs are approximate. They are provided as guidelines for judging roughly where spacecraft are likely to be with many km uncertainty at epoch, and more when propagating to other times. They have never been more than that, and this has been stated time and again.
Probably not. This kind of noise, at the level of 700 meters in SPG4 propagated orbits, probably happens all the time.
See the discussion in this answer; it seems to be the same exact phenomenon. There is a "blipping" of the eccentricity, but not a change in the semimajor axis, so the periapsis goes one way, and the apoapsis goes the opposite way, then everything goes back to the way it was before the blip (except of course for the steady decay in this case). The explanation is probably the same as well; a small error in position can always happen, especially if at a point in the orbit where there are statistical correlations between fitted parameters. As you point out, the positional error is only several hundred meters.
An error that large would be much less likely in the semimajor axis, because this is very strongly linked to mean motion. As your data the data for the object shown below both show, that doesn't happen. A mis-measurement of a few hundred meters than change the fitted eccentricity easily, but since satellites are observed over and over, timing and phasing but a much stronger lock on the semimajor axis.
I'll bet if one were to start looking, noise and blips in eccentricity is probably fairly common. It's likely a non-event. EDIT: I started looking, and sure enough, it's quite noisy!
I'll add some more about ellipticity noise in a few minutes...
below: This plot from here shows a much larger blip in eccentricity for another spacecraft In this case it's maybe 20x larger than that shown in the question. By coincidence, in this plot Tiangong-1 was used as a reference, and does not show significant noise at this scale.
This is data from about 01-oct-2017 to 31-dec-2017:
below: This is only the Tiangong-1 data, showing the change in peri, apo, semi (top, scale = km) and eccentricity (bottom) from one TLE to the next, over the same three months.
This shows that noise at the 1km scale for Tiangong-1 is commonplace.