The answer to both questions is “yes.” The facilities Israel had can be called either toilets or latrines.
Latrine is the definition of aphedron in the standard lexicon of New Testament Greek (BDAG). Latrine perhaps is the most technically correct word for what Jesus is talking about because it is the most comprehensive. A latrine can be a ditch or hole in the ground, a structure similar to an outhouse, or a nice public rest room constructed of stone with constantly running water—the Jews had all these. The most famous latrines of Jerusalem were those of the Essenes (appropriately located not far from the Dung Gate). Qumran also had a communal latrine.
Toilet is also an appropriate translation. The readers of the gospels were familiar with large public toilet facilities with running water in the larger cities. The most famous one in Israel is at Beth Shan. The most famous of all is the large public toilet facility at Ephesus. (If you google either latrines or ancient toilets, you will find pictures of these.)
If a toilet is a “fixed receptacle for defecation” (dictionary definition) the Jews also had toilets. There are half a dozen stone toilet seats from the destruction layers of Old Testament Jerusalem. The rich and famous did not use outhouses that were outside the city as the law required. If you google “outdoor toilets” you will find pictures of outhouses not very different from the ones Israel would have had, many of them with the word toilet written right on their wall. The word toilet is not limited to our modern flush toilet. So, yes, it is correct to say the Jews of Jesus’ day had toilets.
The Jews had outhouses, holes in the ground, public latrines that were pit toilets, public toilets with running water, and sewers. Though either latrine or toilet would be a correct translation of what Jesus was talking about, we will probably use latrine because it is the more comprehensive term and some readers might have the mistaken idea that people in Jesus’ day did not have toilets. The problem with latrine is that in modern usage it often is limited to military latrines. Another reason to use latrine is that it will also work in those Old Testament passages which deal with latrines and indoor toilets.
The other choice that must be made is between some sort of word for toilet or a euphemism about passing out of the body. Jesus could have said “out of the body” but he did not. Jesus chose a more jarring word. The reference to a toilet or latrine was even more jarring in Jewish culture because the latrine was an unclean place that was supposed to be outside the settlement. Jesus’ choice of a more jarring term makes his point about the folly of judging people by what kind of food they put into their mouth all the more striking.
Over all, although it has something of an archaic sound, latrine is probably the best choice. Also translators should not censor Jesus by hiding his choice of words.