If 9 people ate bread and one had vegetables, they make a zimun with Hashem’s name. Same with 8 or 7 who ate bread and 2 or 3 ate vegetable. However, if 6 ate bread and 4 had vegetables, it is a dispute in the gemara. One opinion says there is zimun of 10 bc a majority ate bread, but a second opinion says there’s no zimun of 10 bc 6 is not a discernible majority.

After King Yannai murdered all the Rabbis he was eating a meal end no one knew how to bentsh. His wife, Shalomtzion, produced her brother Shimon Ben Shetach to lead bentching. (She’d been hiding him previously) They brought him a cup of wine for bentching but he refused to lead bentching until he had eaten, so he drank the cup of wine first before bentching and then led bentching. However, the Gemara disapproves of this, bc even though we said that a zimun of 10 can be completed by people who don’t eat bread, the one who actually leads birchas Hamazon Jsut have eaten at least a cazayis or bread.

First bracha of Birchas Hamazon was instituted by Moshe when the Jews ate the manna. Second Bracha is from Yehoshua when the Jews entered Israel and the 3rd bracha is a combination from dovid and shlomo with Jerusalem and the Beis HaMikdash. The fourth blessing was added after the Beitar revolt when the Jewish bodies were allowed to be buried and they miraculously had not rotted.

It is a dispute between the tannaim if zimun preface to bentching has a source from the Torah or not. (וברכת את ה אלקיך) There’s also a a dispute if the fourth bracha is from the Torah.

What source to say hamotzi before eating? Some say it’s a kal vachomer logic that if we thank HaShem when were full, certainly we should thank Him when we’re hungry. Others bring various textual sources for blessing before bread.

Source for birchas hatorah acc to one opinion is within birchas Hamazon itself (הארץ הטובה)

To fulfill birchas Hamazon, one should metion bris and Torah in the 2nd bracha and Kingship of Dovid in the 3rd bracha.