What is the "Jews' language" spoken of in Nehemiah 13:24?

A modern translation of the passage in question, Neh 13:23-29, clears up your 3rd question: "half" referred to half of their children, who could only speak the language of their wives, the language spoken in Ashdod, Ammon, or Moab.

Here's verses 23-24, ESV translation:

²³ In those days also I saw the Jews who had married women of Ashdod, Ammon, and Moab. ²⁴ And half of their children spoke the language of Ashdod, and they could not speak the language of Judah, but only the language of each people.

Your first two questions can be answered from the following excerpt from the New International Commentary of the Old Testament: The Books of Ezra and Nehemiah. In short, "speech of Ashdod" refers to a language very similar to Aramaic spoken by the children, while "Jews' language" refers to the Hebrew language used in the temple.

23-27 Another continual problem was the marriages to foreigners (cf. Ezra 9–10; Neh. 6:18; 10:31 [Eng. 30]). On his arrival, Ezra had tackled this problem. Drastic measures were taken and the foreign wives and their children were sent away. But when Nehemiah came, the problem was still not resolved, as we learn from Neh. 6:18. As part of the covenant renewal described in Neh. 10 the marriages are again mentioned. The people solemnly undertook not to marry foreigners. In Neh. 13:1–3 the presence of Ammonites and Moabites in the cultic congregation is strictly forbidden. So after Nehemiah’s departure to the Persian court this problem must have developed again. If we take the view that everything was under control when Nehemiah left Jerusalem, we may deduce from the fact that the children spoke a foreign dialect that quite a time had elapsed between his departure and arrival. Soon after his arrival he discovered that children of foreign wives of Jews spoke Ashdodite, Ammonite, and Moabite. We know from inscriptions that Moabite was closely related to Hebrew.¹ We may presume that Ashdodite and Ammonite were languages very similar to Aramaic.² This is substantiated to a certain extent by the discovery of an ostracon in Nebi Yunus that is written in Aramaic.³

Ashdodite: for Ashdodites cf. our discussion of 4:7:

The Ashdodites were the western neighbors. Their territory consisted of the whole area where the Philistines once lived.³ To the north were the Samaritans with their leader Sanballat. From a bird’s-eye view Nehemiah gives the four neighboring provinces of the Persian empire. The plot against Nehemiah was contrived by all his neighbors, a grave situation indeed. Judah was totally isolated, cut off from the caravan routes from South Arabia and from the Mediterranean Sea to the west.

The language of the Jews (Heb. yehûḏîṯ): this is quite probably Hebrew. If we take all this into consideration, we may presume that there was little linguistic difference between Hebrew and Moabite. The vernacular of the Jews was probably Aramaic, so that some difference between this and Ashdodite and Ammonite would exist. What was then precisely the difference? No doubt dialectal differences were involved as well as the different pronunciation of sounds.⁴ Nehemiah was able to distinguish between the language spoken by children of foreign women and by regular Jewish children. Because Hebrew was the language of the cult, people with such a different language could not be permitted to participate in Jewish religious life. For the religion it was therefore a new and dangerous development. Nehemiah tackled this problem vehemently. He brought in a court case (rîḇ) against his people before the Lord (cf. 5:7; 13:11, 17).

As to whether the Hebrew language was Biblical Hebrew (BH) or Mishanic Hebrew (MH), see quotes from the paper A Linguistic Introduction to the Origins and Characteristics of Early Misnaic Hebrew as it Relates to Biblical Hebrew):

Mishnaic Hebrew (MH), also referred to as Rabbinic Hebrew (RH), characterizes Hebrew literature produced by rabbinic scholars from approximately 70 C.E. to 400-500 C.E. (thus, the common phrase leshon hakhamim “the language of the sages”). Within this timeframe, MH can be divided into the earlier language of the tannaim “repeaters” (ca. 70-250 C.E.) and the latter language of the amoraim “speakers” (ca. 3rd-5th century C.E.). Tannaitic Hebrew is found in the Mishna, Tosefta, Halakhic Midrashim, and Seder Olam Rabbah, while Amoraic Hebrew characterizes the Jerusalem Talmud, Haggadic Midrashim, and the Babylonian Talmud.

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The notion that MH existed orally for centuries prior to the Tannaitic period is generally accepted, but what was the process by which it became a literary language? Kutscher argues that a Hebrew-Aramaic mix developed after the exile and was used throughout Palestine until the 1st century B.C.E. when Rome invaded Judea. Instead of MH flowing directly out of BH as Segal argued, it is more likely that both BH and MH developed synchronically as diglossic dialects. If this was indeed the case, BH would have functioned as a “high” but dead literary language and MH the “low” vernacular.

According to Kutscher, foreign attacks on the Jewish state, their political center, and their national identity, served as the impetus to BH phasing out and MH replacing it as the standard literary form. MH then became the new “high” language with Aramaic and Greek becoming the vulgar tongues.

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It appears highly probable that MH did exist as a living language in and around Palestine in the Second Temple Period, and in many ways developed out of and alongside BH. The features of MH highlighted in this study are characteristic of that development and additionally testify to significant Aramaic influence.

Since the process of MH development must barely have started during the time of Nehemiah, the high literary language used in the temple must have been Biblical Hebrew.