Omission of the noun in the second sentence of two sentences joined by a conjunction like "and"

First, I want to make clear that I’m not a native English speaker. So this question may seem too easy for native English speakers, but it is quite difficult to definitively answer for me.

I have seen, in several websites and scientific writing, people omitting the noun in the second sentence, for example:

The boys are playing football, and seem to steadily improve their technique.

The sentences look correct, but my concerns are: What if the sentences are too long, would be still understandable and acceptable to omit the second noun or pronoun?

How about:

The boys are playing football with a courage and discipline that I have never seen before, and seem to steadily improve their technique to handle the ball as well as their team work.

I would prefer to use "They" before "seem" in the second sentence.

Which way will be more proper?

And… Are there any rules as to when to omit the noun of the second sentence or when not to do it?