The Rishonim (Rabbeinu Yehuda Ben Yakar, Rabbeinu Yonah, among others) describe these Kedushos as סיפור דברים, relating what is happening (as opposed to Kedusha D'Amida, in repetition of Shemoneh Esrei, which is our own Kedusha - where we say נקדש, let us be mekadesh). The lines between serve as explaining what is happening - they give the context to who says those lines and how they say it.
In terms of the congregation saying them out loud
Kedushas Yotzer: the Rama in O.C. 59:3 says the congregation should say the lines of Kedusha out loud.
Kedushas Uva Letzion: Rama in O.C. 132:1 says that the Aramaic should be said silently, but the implication is that the verses have the same rules as by Kedushas Yotzer. The Avudarham implies that the final verse, Hashem yimloch, should be said quietly.
In terms of skipping ahead to say it:
the Mishna Berura in 132:3 says regarding Kedushas Uva Letzion that if you have not yet said Ashrei, you should skip ahead to respond with the congregation. In Shaar Hatziun he cites Magen Avraham that even before you have davened you should skip ahead, but he concludes that this is not the minhag.
Regarding Kedushas Yotzer, it is not explicit, but it would seem to me that it depends on whether or not you will say it by your self when you get there - the Mechaber O.C. 59:3 brings two opinions as to whether an individual says Kedushas Yotzer on his own, and the Rama concludes that he does. The Vilna Gaon says that he shouldn't. The Gemara in Berachos 21b has a dispute if one comes into the synagogue and finds the congregation already davening, whether he should start davening if he will not be finished in time to say Kedusha with the congregation:
אמר רב הונא הנכנס לבית הכנסת ומצא צבור שמתפללין אם יכול להתחיל ולגמור עד שלא יגיע ש"ץ למודים יתפלל ואם לאו אל יתפלל ריב"ל אמר אם יכול להתחיל ולגמור עד שלא יגיע ש"צ לקדושה יתפלל ואם לאו אל יתפלל במאי קא מפלגי מר סבר יחיד אומר קדושה ומר סבר אין יחיד אומר קדושה
Rav Huna holds he should start, and doesn't have to worry about missing Kedusha with the congregation because he will say it on his own. R' Yehoshua ben Levi holds that you should not start if you won't be able to say Kedusha with the congregation, because he holds you will not say it on your own. But both agree that would you be saying it on your own, you wouldn't need to go out of your way to ensure saying it with the congregations. So it seems that if you will say it on your own, you need not worry about saying it with the congregation.
Therefore, I would have assumed the Rama would tell you not to jump ahead, and the Vilna Gaon would tell you to jump ahead.
However, the same logic could have applied by Kedushas Uva Letzion, and yet it is brought there without qualification that you should skip ahead.