One word to a full sentence can be the subject of a sentence:
"Das ist ekelhaft!" als Ausschrei ist eine Äußerung des Ekels!
However, can I also chain two sentences in one quote and use them as the subject?
"Das ist ekelhaft! Und ich hasse dich!" als Ausschrei ist eine Äußerung des Ekels!
Note that the purpose I am going for is a konjunktionalphrase with als, and I am not interested in indirect speech and do not intend to chain two separate quotes with conjunctions like und.
You did it right.
But the noun Ausschrei does not exist and you can't use das Ausschreien either because it does not mean to cry out in an intransitive sense. Rather, it means the same as jemanden ausschimpfen but more loud and you have to name that jemand.
I recommend der Aufschrei instead, or das Gekreisch.
And mind the quotes. Those have to be low 99, high 66 in German.
You would actually use the verb and "Äußerungen" in plural, and you would put the "und" outside the quotes. Janka already told you to use "Aufschrei" or, alternatively "Ausruf" instead of "Ausschrei", so:
"Das ist ekelhaft!" und "Ich hasse dich!" als Aufschreie sind Äußerungen des Ekels.
