Does Amsterdam offer free museum admissions on the first weekend of the month like some other cities? Any advice on any other opportunities for free admission, such as late in the day etc?
No:( Look into the IAmsterdam card, which includes a unlimited transit pass. You'll have to do the math to see if it's cost effective. It doesn't cover Anne Frank or the Rijks, but generally if your visit three covered attractions in one day, and will be making at least four trips a day on the trams it will save you money: http://tinyurl.com/agatl2f
If you're seeing a lot of museums, the Netherlands Museum Card can be a good deal. It's about 45 euros, is good for a whole year, and covers places all over the country.