As always, read with a gemara in front of you. Or, you could just skip it, of course.
The gemara’s contention that Gd protects the righteous from accidental sin has its roots in various sources, such as the promise that HaShem will protect a Sanhedrin from accidentally killing an innocent victim. This is problematic, though, for we find cases of rabbinic error; whole segments of gemara deal, for example, with judicial error. Tosafot השתא tackles the problem by distinguishing between eating non-kosher and other sins; see his comments there.
דום may be taken as “be silent” or as “hope for.” But see Rosh haShanah 16b, as well as teshuvah #6 of the Tzemach Tzedek (the first, not the Lubavitcher Tzemach Tzedek), on the issue of liability for the causing the downfall of others.
Note that our gemara mis-cites Hosheia 9:1; the word there is כעמים, not בעמים. This may be a typo, but it may also be an אל תקרי approach, since the adjusted meaning of the sentence more closely fits our gemara’s point.
See Tosafot זמרא on the permissibility of listening to music in our own day. Ashkenazi poskim, like Tosafot, seem to be more lenient than Sephardic poskim in this issue.
See Tosafot אף על גב on the point that settling Israel justifies אמירה לעכו"מ, but other mitzvot do not.
See Tosafot הדר.
Tosafot שוו raises a very interesting question: We want to aid proper verification of a get in order to help a woman re-marry, but where is our incentive to make it easier to verify a document freeing an עבד? In the Torah’s version of עבדות, which is primarily long-term economic commitment, where is the harm? If he cannot marry a בת חורין, he can marry a שפחה - and the gemara later, on 13a, will contend that he prefers this! Tosafot offers a suggestion which runs counter to 13a, and an additional suggestion that the עבד wishes to fulfill mitzvot. Ramban and Ritva, though, will say that the עבד who is in limbo can marry no one, and so our expedited verification will help him a great deal.
Rashi חוץ מגיטי נשים takes the view of the Chachmei Provence (2:48) that the reason secular governments have judicial authority recognized by Jewish law (dina d’malchuta dina) is because Gd instructed them to carry out such laws, in the mitzvot of Bnei Noach. For more on this see my post here.
Rashi בגיטי נשים does not seem to recognize any עיגון situation for an eved - contrary to the remarks of Tosafot שוו on 9a.
If we are to etch in signatures for illiterate witnesses and have them fill the signatures with ink, or if we will use stencils (per Rabbeinu Chananel cited in Tosafot here), how will we recognize their signatures as their own?! Some fifteen years ago, my friend Tzvi Hebel suggested that we don’t have to recognize their signatures at all, just that the delivery agent has to see them “sign” the document, and this is indeed fulfilled.