There are several trusted international organizations that can provide reports and summaries about LGBTI rights in various countries, as well as international law.
1) The International Lesbian, Bisexual, Gay, Trans and Intersex Association’s website allows you to search by country: https://ilga.org/civi_details
2) Human Rights Watch has information about different countries that you can search. Go to the bottom of the homepage to “Browse by Region”: http://www.hrw.org/
3) Amnesty International allows you to search for human rights information by both country and topic on the right side of their homepage: http://amnesty.org/
4) The Outright International provides information by country: https://outrightinternational.org/where-we-work