What Torah commandments may a Noahide do?

[Director Michael]

What is the ruling regarding a Noahide family building and using a sukkah during Sukkot?

See the explanation in The Divine Code, Part I, topic 3:6 and the footnotes there. That chapter is posted on our web site, together with chapters 1 and 2:
https://asknoah.org/wp-content/uploads/f...ers1-3.pdf

I'll give you some additional explanation here.

Where Rambam says in Hilchot Melachim 10:10, "We should not prevent a Gentile who desires to perform one of the Torah’s mitzvot in order to receive reward from doing so," he means an actual reward (i.e. benefit), not an incorrectly imagined reward or benefit. A Gentile does not receive any actual reward or benefit for performing a ritual religious observance that G-d uniquely commanded upon the Jews.

Here is an example of one of the false arguments that is offered to try to contradict this. They partially quote the Midrash Kohelet Rabbah 1:9, which starts by speaking about not eating pork (a ritual observance commanded to the Jews). The first sentence says:

"The Rabbis say: In the future, the Holy One blessed be He will dispatch a herald and proclaim and say: ‘Anyone who has never eaten pig meat, let him come and collect his reward.’ Many of the nations of the world who never ate pig meat will come to collect their [spiritual] reward."

But in the very next sentence of that midrash, G-d goes on to say to those Gentiles that they do not receive any spiritual reward for that, and G-d criticizes them for expecting to receive a reward. It says there,

"At that moment, the Holy One blessed be He will say: ‘Shall these [Gentiles] be rewarded in two worlds [physical and spiritual]? Is it not enough that they enjoyed [physical benefits in] this world, but they seek to enjoy the [spiritual] world of My children [the Jews], too?’ At that moment, the Holy One blessed be He will dispatch a herald a second time, and proclaim and say: ‘Anyone who has not eaten the meat of animal carcasses and mauled animals [those terms in Hebrew refer to different categories of meat that was not of a kosher animal that was correctly kosher-slaughtered for the sake of Jewish consumption], repugnant creatures [e.g., tiny insects or insect eggs hidden on or in fruits, vegetables and grains which only Jews are commanded to check for, and people generally are not aware of them] and creeping animals, if he did not eat of his own, he [definitely] ate another’s [who did not specifically avoid or check for such things].'

The intent of the midrash is that only Jews are commanded about those ritual dietary laws, and only Jews will get any spiritual reward for observing them, and Gentiles / Noahides should not expect or think that they will get any reward for observing them.

In the very same paragraph of Hilchot Melachim 10:10, Rambam goes on in the next two sentences to give the best examples from the two specific categories of Jewish mitzvah observances that Gentiles do receive actual spiritual and/or physical benefit from, and which they are therefore permitted to observe. Quoting Rambam:

1.  "If he brings an animal to be sacrificed as a burnt offering, we should receive it."

Sacrificing an animal to G-d as a completely burnt offering is a purely spiritual observance, with no logical physical benefit for the person, but it is not a uniquely Jewish observance. Torah teaches that Adam, Abel, Noah and Naaman all permissibly brought burnt-offering sacrifices to G-d. So this is permitted to Gentiles, and therefore if they want to do it themselves outside the Holy Temple, but according to the method of slaughter and burning that is commanded for the Jews, they may study and be instructed about how to do so.

However, in our time, sacrificing animals to G-d is not practiced and we don't encourage Gentiles / Noahides to do that. But it serves as an example of spiritual practices that are not uniquely commanded to Jews, that are permitted for Gentiles / Noahides. For example:

-  Prayer to G-d. We see that Cain prayed to G-d for mercy. Eliezer prayed to G-d for success in his mission to find a wife for Isaac, and he praised and thanked G-d when he was successful.
-  Male circumcision for the sake of G-d, as explained in The Divine Code, Part I, chapter 3. (Note that it can have a spiritual aspect for Gentiles as explained there, and it has a practical physical benefit.)

     (b) "If a Gentile gives charity, we should accept it from him."

Giving proper charity bestows a practical actual benefit to the recipient of the charity, and G-d actually obligates Gentiles to do this. G-d's decision to destroy the metropolis of Sodom was sealed because they refused to give charity, and they even made it illegal in their society.
Furthermore, in the Talmud it states that Gentiles receive a spiritual benefit from giving charity. So Noahides may give charity in accordance with the Jewish Torah laws for doing so.

As explained in The Divine Code, Part I, chapter 3, this extends to permission for Gentiles to practice (in a general way) other of the Jewish mitzvot that provide practical, actual physical benefits for other people (e.g., returning lost objects, getting married, having children), or practical benefits for other creatures (e.g. kosher-style slaughter because it is virtually painless to the animal), and even practical benefits in terms of moral refinement of a person's personality (e.g. not taking revenge and not holding a grudge).

In regard to marking the significance of the seven days of Sukkot, you have options that apply in general for the Jewish Sabbath and festival days (including the the days of Sukkot  that are Yom Tov days). See the answers to Question 1 this page of our web site, about permissible activities on the Jewish Sabbath, which also apply to the Jewish Yom Tov days: https://asknoah.org/essay/the-seventh-day

Here are some things that you could do during Sukkot. All of them are normal things that people do all year long, and children enjoy them, so your family can do any of them during Sukkot:
-  Eat your meals outside, in a comfortable place when the weather is nice. This could include picnicking.
-  Take one or more family day trips or multi-day trips, and you'll be spending more time out of your house.
-  Go camping.

All of those normal activities can help to reinforce the knowledge that we are really only guests in this world, passing through temporarily by G-d's grace and kindness. And that we should not rely on our physical trappings (including the walls and roof of our homes) for protection, but rather our protection comes from G-d Who provides those for us as long as it is His will to do so. And therefore we should put our faith and trust only in Him.

There are also special Chumash and Haftarah (Prophets) sections for Sukkot that you can read. You can also read the book of Kohelet (Ecclesiastes) which is customary for Sukkot. Noahides may read those along with the straightforward (peshat) traditional Rabbinical explanations, most preferably the explanations by Rashi.