A dead body transmits tumah to any sized material directly above it. There’s another law that when there is a “roof” above a dead body, at least one tefach thick, then the tumah spreads to anything under the same roof. By rabbinic law, one of the 18 decrees, a roof can be even something that is 1/3 of a tefach.

There is a law in tumah for foods that it cannot accept tumah unless it first became wet with a liquid. The law is that the emergence of a liquid has to be לרצון- acc to the will of the owner. If someone harvests grapes in a basket this has small holes and the liquid can seep out, presumably it is not to the will of the owner and it therefore cannot make the grapes susceptible to tumah. That was the opinion of Hillel. However, Shammai decreed as one of the 18 decrees that the juice from the grapes is “machshir.” One reason is that we’re concerned the basket may be impure and the liquids will become tamei. Tamid liquids are considered significant bc of their tumah and so they would be able to be “machshir.” A second reason is we’re concerned for a basket lined with pitch that would contain the liquid and it would be in the will of the owner. A third reason is we’re worried for a clusters of grapes that get tangled and the only way to untangle them would be to press them and causing liquid to come out (so it’s considered to the will of the owner) A fourth concern is one may check on the ripeness of the grapes by pressing them and causing liquid to emerge to his will.

By the letter of the law, if one replants Terumah in the ground, what emerges new is ordinary chulin. However, one of the 18 decrees is that the growth from Terumah is considered Terumah. The reason is we’re concerned a Kohen may have Terumah that became tamei, and instead of burning it like he should, he would wait for the planting season to replant it. (We don’t want tamei Terumah just sitting around bc someone may eat it)

If someone has their wallet on them right as Shabbos is starting and they are on the road, they should instruct a non Jew to carry it in for them. This is preferable over walking less than 4 Amos at a time in the street. (The rabbis have this leniency bc or else ppl would jsut carry it outright)

More of 18 decrees is to prevent intermarriage, the rabbis forbade bread and oil of non Jews. This is to prevent us from drinking the wine of non Jews, which would lead to intermarriage and then indolatry. Moreover, the rabbis said a non Jewish boy is tamei zav so we won’t allow our children to hang around them ( it could lead to homosexuality)

Beis Shammai holds your vessels have to rest on Shabbos, so one can’t soak inks or dyes before Shabbos unless it will be completely soaked before Shabbos begins. Beis Hillel holds ones vessels can work for them on Shabbos so this is permitted. Another illustration of this would be setting a trap for animals b4 Shabbos.

Beis Shammai does not allow one to do an action that can be misinterpreted as a Jew instructing a non Jew to do work on Shabbos but Beis Hillel permits

Beis Shammai does not allow one to give laundry to non Jew unless there is enough time for the laundry to get done before Shabbos.