Applying for international graduate school without any prior collaboration with researchers at the new location
There are a huge number of questions on this site about how to contact potential PhD supervisors. Generally the advice is:

Before contacting anyone, make sure you understand how PhD admissions typically work in the country you are interested in. Who makes the decision? Do you apply for a specific project, or for a generic graduate programme? Where does the money come from?
Be brief. Professors are busy people. Don't give them a wall of text and 17 attachments to read.
Have clear questions, and ask them clearly. Good starting points are: (i) Are you interested in taking on a student to start in September 2023 [optional: in the general field of XYZ]? (ii) If so, what kind of topic would you have in mind? (iii) If I were to apply for funding from XYZ, would you be willing to support that application? Are you aware of any other sources of funding that might be relevant?

Note that you don't have to put absolutely everything into your first email. If someone is genuinely interested in your application, they will be happy to answer one or two sensible follow-up emails. Also note that professors get lots of these emails - some students even (appear to) use automailers to send them to everyone in the world -  and, statistically, most of those people will need to be declined.
Your question mentions specific aspects of support that you may require. I suggest:

Financial support is easy - just make it clear in your email that you have not yet obtained a scholarship.
Immigration - Mention in your email what citizenship(s) you hold; it is often relevant for identifying which sources of funding may be available.
'Psychological' support (I wonder if 'emotional' support may be a better phrase here) - Unless you have a concrete requirement that you need potential advisors to accept, this is probably not something that needs to be addressed in an initial email. Once you have established that the professor is interested in you - and you are interested in what they can offer - you can ask more detailed questions about life in the city/university/etc. This is a good opportunity to gauge their personality and advising style. It is also very normal to ask to be put in touch with current students, who are often better-placed to explain what the experience is like 'on the ground'. Many universities will have societies/associations focussing on LGBT and international students, and these will be able to put you in touch with someone who can address any more specific questions you have.