The notes here are taken from the actual Scala, so be warned that references to the "previous" proverb refer to its order in the Scala, not its order here. You can read more about the word at the Verbosum blog: DOMUS.
Omnis est rex in domo sua. ~ Note: Compare the English saying, "A man's home is his castle."
Dulce domum. ~ Note: Notice the use of the neuter adjective here: the proverb does not say that a home is sweet, but that home is a sweet thing, a pleasant thing. This is also the title of the school song of the Winchester College, a boys' school in Winchester, England.
Romani, ite domum!
Stet fortuna domus! ~ Note: Careful with domus here: it is the genitive singular, so fortuna domus means "the good fortune of this house." Note also the subjunctive: stet. You can also find the saying in this form: Stat fortuna domus virtute.
Parva domus, parva cura. ~ Note: Here a parva domus does not have any negative connotations at all; this is small in the sense of economical, modest, etc.
Domi manere convenit felicibus. ~ Note: This expands on the previous proverb, letting us know just who should stay at home: those who are happy, felicibus. To go out would just be to put your happiness at risk! Compare the saying in the Adagia of Erasmus, 3.1.13: Domi manere oportet belle fortunatum.
Domi manendum. ~ Note: This is one of the sayings Erasmus included in his Adagia, 3.9.44. Domi is the locative form of the word domus, meaning "(at) home." The gerundive is being used impersonally here with the force of a command: "stay at home."
Domus propria, domus optima. ~ Note: This is a variation on the "cuique suum" theme, this time using the adjective proprius to express that idea of "one's own."
Domus amica, domus optima. ~ Note: This Latin saying (included by Erasmus in his Adagia, 3.10.38) has something in common with English saying such as "Home, sweet home" and also "Be it ever so humble, there's no place like home." As you can see here, amicus is actually an adjective in Latin, "friendly," although you are far more likely to find it used substantively, "friend."
Propria domus omnium optima. ~ Note: This version of the same proverb spells out the superlative comparison explicitly: one's own home is the best (optima) of all (omnium).
Domi manere tutum. ~ Note: The form domi is an archaic locative, "at home." The infinitive manere is regarded as a neuter noun, hence the neuter form tutum: Staying at home is safe.
Domus divisa contra se non stabit. ~ Note: The words are from the Gospel of Matthew, 12.
Quaere vicinum ante domum, et socium ante viam.
Domi suae quilibet rex. ~ Note: Compare the saying you saw earlier: Omnis est rex in domo sua.
Luctu vacat bos cum senex moritur domi.
Una domus non alit duos canes. ~ Note: This is one of the sayings collected by Erasmus in his Adagia, 2.2.24.
Canis domi ferocissimus.
Parva domus, magna quies. ~ Note: This is a nice paradoxical parallel: parva/magna, domus/quies.
Qui altam facit domum, quaerit suam ruinam. ~ Note: This saying is included by Polydorus in his Adagia, B321.
Beatus is qui in prosperis manet domi.
Domi manendum est, fata cui sunt prospera.
Noli esse sicut leo in domo tua, opprimens subiectos tibi.
Nec domum esse hoc corpus, sed hospitium.
Domum tuam guberna. ~ Note: You can sometimes find this saying attributed to Chilon of Sparta, one of the legendary "Seven Sages."
Nusquam commodius vivitur quam domi.
Exsul ubi nusquam domus est, sine sepulcro est mortuus.
Domus pulchra dominis aedificatur, non muribus.
Domi leones, foras vulpes.
Domi leones, in pugna vulpes.
Totidem domi hostes habemus quot servos. ~ Note: Compare the saying in the Adagia of Erasmus, 2.3.31: Quot servos habemus, totidem habemus hostes.
Nulla pusilla domus, quae amicos multos capit.
Serpens eiciendus e domo.
O domus antiqua heu quam dispari domino disparis.
Tutius est domi, quam alibi, manere.
Foris sapere, domi desipere.
Non domus est pacis, ubi regnat lingua loquacis. ~ Note: This is one of the sayings collected by Wegeler, 721.
In domo Patris mei mansiones multae sunt. ~ Note: This saying is included by Polydorus in his Adagia, B145.
Domi manere virum fortunatum decet.
Domi talpa, foris lynx.
Domi talpa, foris Argus.
Ex domo felis discedit mus impransus.
Melior est buccella sicca cum gaudio, quam domus plena victimis cum iurgio.
Qui conturbat domum suam, possidebit ventos.
Fur male furatur, cum fur domui dominatur.
Dominus est assiduus servus suae domus.
Pax huic domui, et omnibus habitantibus in ea.