I saw a Jewish scholar (Jacob Emden) claim that according to Islam, when a man dies without children then it is forbidden for his brother to marry the widow, but if he dies with children, then the living brother is supposed to marry his dead brother's widow. Is this true? (This is the exact opposite of the Jewish levirate marriage whereby if the deceased brother was childless, the living brother must marry the widow, and if the deceased brother was not childless, he is forbidden from marrying her.) I saw https://en.wikipedia.org/wiki/Levirate_marriage#Islam but this doesn't give me enough information.
No this is not true. It is possible that it is a cultural practice somewhere but is not based in Islamic law.
A person is neither forbidden from marrying the widow of his brother nor is he obligated to marry her, regardless of the presence of children. Rather it is permissible - they may marry if they want to and are neither forbidden nor encouraged to do so.
The wife of one’s brother is not unmarriageable to him forever; rather, as soon as the marriage between her and her husband ends through divorce or death and her ‘Iddah (waiting period) lapses, it becomes permissible for the brother of her ex-husband to marry her.
The only people a person is forbidden to marry are his\her mahrams. See Who are the women that a man is forbidden to marry? The relations of the husband who are forbidden to be married by the wife after his death are the husbands's father and children - not brothers:
ولا تنكحوا ما نكح آباؤكم من النساء ... وحلائل أبنائكم الذين من أصلابكم
And do not marry those [women] whom your fathers married ... and [also prohibited are] the wives of your sons who are from your [own] loins
وقل للمؤمنات ... لا يبدين زينتهن إلا ... آباء بعولتهن ... أو أبناء بعولتهن
And tell the believing women to ... not expose their adornment except to ... their husbands' fathers ... their husbands' sons
There isn't any obligation on a person to marry someone specific. Rather it is permitted to marry whomever one likes among the permitted women:
فانكحوا ما طاب لكم من النساء
then marry those that please you of [other] women
The widow is free to marry anyone or even to remain unmarried after the death of her husband. There is no restriction upon her, except for observing the mourning period:
فلا جناح عليكم فيما فعلن في أنفسهن بالمعروف
then there is no blame upon you for what they do with themselves in an acceptable manner.
A person does not have the first right to his deceased brother's widows. That was an ancient Arabian custom that the Quran forbade:
لا يحل لكم أن ترثوا النساء كرها
it is not lawful for you to inherit women by compulsion
كانوا إذا مات الرجل كان أولياؤه أحق بامرأته: إن شاء بعضهم تزوجها، وإن شاءوا زوجها، وإن شاءوا لم يزوجها، فهم أحق بها من أهلها، فنزلت هذه الآية في ذلك
The custom (in the Pre-lslamic Period) was that if a man died, his relatives used to have the right to inherit his wife, and if one of them wished, he could marry her, or they could marry her to somebody else, or prevent her from marrying if they wished, for they had more right to dispose of her than her own relatives. Therefore this Verse was revealed concerning this matter.