Is "další" the masculine form of another? Is there ever a time when it would become dalšá? Why does Katerina get a masculine form (if that is what další actually is)?
"Další" follows "jarní" pattern, thus it doesn't change.
I got dalsi underlined but I used the accent in the right spot?
You might have forgotten to add the acute accent to the i? (i.e. další is correct, while dalši is not).
It can be hard to see the difference on some screens.
it accepts jiná kateřina as well as další Kateřina, what is the distinction?
'jiný' = different, another, other...
'další' = next, another, one more, ...
If both mean another then, which is more formal? Which one would be used in the sentence- Not that Katerina but another Katerina.
This has nothing to do with formality.
The English word another has two meanings, next or different. In Czech you have to choose one of these.
IMO, your suggested sentence would be more likely to use jiná: "Not THAT Kateřina, but a DIFFERENT Kateřina." But další might also be used, depending on the specific context. [Edited to correct my own mis-typing when initially replying... :-/ ]
Why not druhá Katerina?
Druhý is an ordinal number: 'second.' Not really applicable here, I think.