In images of sun's corona there are clear bright lines - nearly 1D stable structures interpreted as magnetic flux ropes, often suggested to have a topological nature...
What is shown in your first image are called coronal loops. They are not one dimensional and they are not really stable. Well they can be stable on time scales of hours or even days, but plasmas like that found in the corona are inherently unstable.
What is actually shown in this image is an iron emission line (I don't know which as I don't know the actual source of the image). The apparent thickness of these loops are actually strongly dependent upon the angular resolution of the imaging telescope. That is, the first images from Yohkoh and those followed by TRACE showed these nice, beautiful arcades/loops. However, more recent missions with much higher angular resolution telescopes noticed an oddity that still seems a little confusing (at least the last time I bugged a solar physicist about this they said it wasn't fully resolved). The thickness of these loops vs height from the their magnetic footpoints (i.e., where they appear to "touch" a surface in this wavelength/frequency of light) is much less variable than a flux rope should be. In fact, the thickness decreased proportionally with the increase in angular resolution of the space telescopes.
They have some energy density per length, and for example magnetic reconnections shorten them...
That's not really how magnetic reconnection works. For instance, see my answer at https://physics.stackexchange.com/a/559759/59023.
I wanted to ask why these 1D magnetic flux ropes are stable - don't just dissipate?
The magnetic fields in the solar corona are rather large and the plasma in the photosphere is dense enough that the magnetic fields are said to be frozen-in to the plasma (e.g., see https://physics.stackexchange.com/a/452325/59023 for more details). It takes a lot of energy to move a lot of mass and increase tension on magnetic field lines, so things tend to evolve slowly (on impatient human time scales).
What you are looking at are loops that are hundreds of times the size of Earth, i.e., things can take a while to change on such large scales. There are a lot of movies from these spacecraft that have frame rates greatly increased from real time. For instance, the Solar Dynamics Observatory (SDO) takes 10 images in 10 seconds for 10 different wavelengths/frequencies, i.e., one image every ~10 seconds at a single wavelength/frequency. If you concatenate those images together and play them at a 30 frame per second rate, you'd be seeing nature evolve much faster than reality. That is, in one second 30 frames would pass by corresponding to ~300 seconds of real time, i.e., the movies are playing 300 times faster than reality.
So why don't the loops dissipate?
Dissipate into what? The loop is an indirect image of the magnetic field topology in the corona. They appear as distinct, quasi-stable phenomena but if you view in another wavelength the shape, location, and size can change. The telescopes that generate such images have narrow filters, so things can appear to move in the movies but it's very difficult to separate movement from changes in emission. That is, a region can appear brighter precisely because it's closer to the center of the frequency filter for that specific iron emission line. They try to account for the gain/response of the filter bands, but it's still an issue with such data.
The intensity also does not directly correspond to temperature. That is, seeing one of these loops become brighter does not necessarily mean that loop became hotter. It could imply there was merely more emission at the same wavelength/frequency or the emission had a higher gain/response in that filter. Multiple filters and additional information are necessary to determine actual motion of plasma and whether heating actually occurs.
There are also these well known fluxons/Abrikosov vortices in superconductor... Is it due to topological reasons similar as for fluxons?
Yes, kind of. The magnetic field is frozen in to the plasma (see my comments above) and the plasma does have a really large conductivity, but it is not like a type-II superconductor in the sense I think you are going for.
There are models of the solar corona where a fluxon-like approach has been implemented, e.g., https://ui.adsabs.harvard.edu/abs/2007JASTP..69..116D/abstract. However, I have not heard much discussion of such things though I tend to work further out than the solar corona, i.e., in the solar wind. So it is possible this is a valid idea, but I am not sure how about the applicability or the directness of the analogy.
