Conversion from version 1.3 to version 2.1 ?

Version 1.3 and 2.1 can coexist on your computer. Just start using 2.1 for future work and keep 1.3 on your computer for older works.

In general, scores that contain many manual adjustments that were performed in order to work around bugs or limitations in 1.3 won't look good in 20 because the defaults are already so much better, and now your adjustments end up being counterproductive. Also, adjustments that depend on the specifics of exactly how things happen to lay out given the current measure widths etc won't necessarily look good if the number of measures on a line changes, and this is likely to happen when changing major versions as the layout algorithms improve.

In your case, it's mostly the changes you made to attempt to put rehearsal marks at the beginning of systems that didn't translate well. MuseScore 2 provides better ways of achieving this effect, probably best to convert your rehearsal marks over by deleting them and re-adding them. Use the Text Style facility to set the default position more appropriately for your purpose. Right aligned, vertical offset 3sp, horizontal -5sp gives almost exactly the result you want with no manual adjustment required at all, meaning it will also be more future-proof.

You also have lines that were attached one note but then dragged to appear to coincide with another. That's never a good idea and does lead to these problems if the measures widths change. Always best to attach the lines to the proper notes so they resize correctly as the measure widths change.

One thing you should definitely do is to print all of the scores to PDF. This way you have a neat, usable reference copy, and if at some stage you are unable to access the .mscz files, and have to remake the score (for further editing) you will be able to see exactly what you are aiming at.