Hi and welcome!
Ignore what the school are recommending. I strongly suspect that they want your son off their books because they cannot be bothered to support him properly and are thinking only of how he will affect their exam results staistics 
. It's awful, but unfortunately it is really really common for schools to seek to get rid of pupils who need extra help, particularly in the run-up to exams. There is even a word for it: "off-rolling", which means the LA or school inappropriately encouraging or putting pressure on parents to home educate.
Having said that, there is still a good chance that home education might be the best choice for your lad. Just be sure to get your advice elsewhere, and not from the school! Other home educators are the best bet here, especially those whose children have some of the same issues as your son. You've come to the right place! If you are on any support groups for parents of transgender children, or children with autism, or children with depression and anxiety, you might also want to post there: home education is fairly popular among all of these groups as school can cause such problems. While there are no reliable staistics, I have the impression that such groups are all disproportionately represented in the home ed community. Another good place to get support and information is your local home ed community. They'll also be able to tell you what the local HE "scene" is like so you may get some sense of how easy it could be for your teen to join in with things.
The ed psych also may not know much about home education. It is surprising how widespread misconceptions are among education professionals, even those who really should know better. For example, some may tell you that children who leave school are running away from their problems and will never learn to cope in wider society. But in fact for many children, removing the major stressor in their lives is a simple and highly effective step in restoring their mental health and they may thrive in a different environment.
In the short run, CB and Tax Credits are not affected by your child's place of education. All families who qualify on financial grounds are eligible while their children are still of Compulsory School Age. For children who are being educated with their age peers (nearly all kids in England and Wales) CSA finishes at the end of Y11. After that, you can continue to claim benefits for young people who are in qualifying full-time education until the day before their 20th birthday. Home education counts, and doesn't have to include formal learning, specific subjects or taking exams. A key caveat, however, is that the home education must have begun before the child's 16th birthday, so this may affect the timing of your decision to start home ed. In the spring of Y11 you will be asked for info about your son's future education. When that happens, I recommend getting advice here or on another national home ed forum: the forms are tricky and aren't written with home education in mind, and if you fill them in wrong you may have your benefits stopped while you appeal the decision. Much better to get it right first time!
It would be useful to talk to local colleges now about what they can offer your son. If he feels up to it, he could even start a course this year (e.g. maths GCSE), as part of his home education. An increasing number of colleges are offering courses specifically for under-16s, and those which don't can still take under-16s on courses alongside older learners. He might (or might not) find college more supportive than school, as well as being a more comfortable environment. Some colleges have very good autism support. You can also start to assess whether your local colleges might be a good fit for the following year, as your son plans, and consider alternatives if not.
I'm sure others will be along soon with more ideas, and I will also post back when I have time to answer some of your other questions.