How should one dispose of Jewish newspapers, newsletters, children's parsha sheets, etc. that contain Divrei Torah? I believe some poskim require/allow it to be double wrapped and then left outside to be picked up -- what is the reasoning behind that view? And is the halachah more stringent if parts of actual pesukim are quoted?
Excerpted from Rabbi Heinemann's guidelines (note that if something has an actual name of G-d, that's a separate set of issues; here we're assuming it's Torah but not the name of G-d):
How should one dispose of Jewish newspapers, newsletters, children's parsha sheets, etc. that contain Divrei Torah?
Newspapers and magazines which contain secular information should not be put into shaimos ... The pages that have Torah may be removed and put into shaimos, or the whole paper may be wrapped in paper and then placed in a plastic cover, kli besoch kli, and placed into the garbage.
(I'd heard similarly from Rabbi Welcher. I think it helps that it's not an entire book/newspaper designed for teaching Torah, just a tiny piece of it. The respect due to Torah-teaching material is different than the prohibition of defacing the name of G-d; thus Torah material shouldn't sit with rotten eggs and coffee grounds, but if you double-bagged it that's okay. I think an RJJ article a while back proposed aggregating this type of sheimos and recycling it.)
Homework and test papers in Limudei Kodesh may be shaimos depending on whether [they are] ...
Paper on which a halacha is printed or written with intention to tell the halacha, a commentary of the Tanach, Mishna, Gemara, Rishonim, and Acharonim or a medrash of Chazal which is intended to explain a posuk or to teach us how to conduct ourselves in mussar or hashkafa.
(See the article for more.)
If I understand the article, I think it depends on the homework sheet. If it contains a large, direct quotation from Tanach, then yes. If the sheet itself teaches something, then yes. Otherwise not.
And is the halachah more stringent if parts of actual pesukim are quoted?
Yes if it's teaching Scripture; no if it's just Biblical paraphrasing to make a point (very common in rabbinic Hebrew).
A paper on which three consecutive words of a posuk from Tanach have been written in one line with the intention of quoting the Tanach (as opposed to a melitzah which is not shaimos). Invitations from organizations and individuals that contain parts of pesukim are shaimos, however, the sentence Od Yishama, as frequently included in wedding invitations, is a melitzah, not intended to explain the posuk.