According to the IRS in this webpage, if your partner is a resident alien, they are a US person and thus a domestic partner. it may not specifically say so in this link but we can make a safe assumption is that he/she are domestic partners.
As far as answering your question on how the IRS forms are to be filled out, we cannot comment without viewing the forms, which is impossible in this forum. You just need to read each question and answer to the best of your ability. It may be necessary to seek advice from a CPA if the questions prove difficult.
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