Sabbath restrictions are for Jews only

[Director Michael]

Concerning mitzvot that are not to be kept by an observant Noahide or Noahide chassid. . . Three of the mitzvot concern the seventh day Sabbath: 1) rest

It is very important to understand that the Torah uses technical Hebrew terms that do not necessarily have a simple correspondence in English. The "rest" which a Jew is commanded for the 7th day is to deliberately refrain from 39 specific creative activities that were required for the construction of the Tabernacle and its vessels, and some specific offshoots of these activities. Most of these don't involve any significant physical exertion. Many types of significant physical exertion are not included in this list, and are therefore allowed for a Jew to do on the 7th day. See: https://asknoah.org/essay/the-seventh-day

A Noahide is certainly allowed to take a physical rest and relaxation time, as long as he's not neglecting some job responsibility that he is under hire or moral responsibility to perform. But physical resting doesn't require refraining from switching on an electric light, writing a few words, having some small object in one's pocket when walking outside, running hot water for a bath or shower, etc.

2) to not leave your house. Can we Noachides choose to not leave our house on Shabbat?

You are not correctly understanding the Torah. The commandment to a Jew of "not leaving one's place" means not travelling (by foot) more than 2000 cubits beyond the boundary of the city, unless a special preparation for this is made before the start of the Sabbath.

3) to keep it holy. Are we able to keep shabbat holy?

No. Only Jews have the commandment to keep the Sabbath "holy" for themselves (literally, separated from the weekdays) by (a) refraining from the 39 specific creative activities that were required for the construction of the Tabernacle and its vessels, and some specific offshoots of these activities, and (b) making ritual prayers of ritual "separation" after the beginning of the 7th day (by saying a prayer and blessing of "Kiddush"/"Sanctification"), and after the end of the 7th day (by saying a prayer and blessing of "Havdalah"/"Separation").

Also concerning the Sabbatical year: although Jews are not obligated to forgive our debts in the Sabbatical year, can we Noahides choose to forgive our debtors in the Sabbatical year?

Your question is very confusing. If one Jew loans money to another Jew, the start of the Sabbatical year automatically cancels the loan (the lending Jew is not allowed to collect the balance of the loan from the Jew who borrowed from him), unless the lender makes a special provision with a Beis Din (Jewish court) before the year starts. This does not apply to loans between Jews and Gentiles or between Gentiles, regardless of which is the lender and which is the borrower. As an act of charity a Gentile can cancel a loan that is owed to him, at any time.

and also refrain from agriculture if we live in Israel?

A Gentile can refrain from agriculture whenever and wherever he wants (as long as he is not under hire to do the work), if it's for a practical reason. But not if he has in mind that he wants to observe one of the ritual commandments for Jews that involve agriculture (for example, the rules of Torah Law that pertain to the Sabbatical years in the Land of Israel).