I want equal time with my daughter but the mother won't let her spend the night. Does biological US father have as much rights as undocumented mother?

Presuming all this is in Nevada (answers vary in different places), there are multiple layers here.  First, under current NV law, the two bio parents have equal rights absent a court order to the contrary.  See here.  Who is on the birth certificate? (if no one as father, a "parentage" action is necesary -- see details here).  Gay/straight is irrelevant to this issue under current law, as is (usually) citizenship/being undocumented.  Was there an agreement in advance?  The law has changed in the past couple of years to protect "intended parents" including in surrogacy situations, and it is potentially possible to have three "parents" involved if the agreements were set up correctly.  Bottom line is that your situation is complicated enough, and has enough twists and turns, that you should consider a consultation with a family law specialist, in this firm or some other, versed in these subjects.  If you wish to make an appointment here, see the information posted here.