Here is a nested search that should work in most databases (might not work in Google Scholar) that is fairly all-encompassing:
(lgbt* OR glbt* OR gay OR lesbian* OR "same sex" OR bisexual* OR queer OR transgender* OR transsex* OR genderqueer)
So, to give an example of what a search might look like using the above for the topic of mental health or substance abuse, it might look like this:
(lgbt* OR glbt* OR gay OR lesbian* OR "same sex" OR bisexual* OR queer OR transgender* OR transsex* OR genderqueer) AND ("mental health" OR "mental illness")
Most databases have a way to limit the date range for your search - look in the Advanced Search options or to the left of your results once you've searched and you'll probably find it.
These database are good to search, regardless of your specific topic:
Also try those below that are relevant to your topic:
Re: more avenues for researching, you'll probably have to explore the scholarly and government/community resources to see if more detailed demographics can be gleaned.
Found an incredibly relevant source for your topic? Do cited reference searches to find researchers who are citing that relevant source - their research might be relevant to you as well, and even more current.