Linking suicide to mental illness is vastly overgeneralising. Yes, being suicidal is a mentall illness, since in the DSM 5 a prerequisite for any mental disorder is it hindering you fullfilling your day-to-day lives (this is why fetishes are not a mental disorder, they only become so as soon as fetishists cannot reach orgasm without involving their fetish). People who are suicidal are depressed, this cuases them not to function. The cause of the depression is irrelevant in the diagnose.
Linking increased suicide rates under transgenders and LGB is correlation, there is no causation whatsoever.
In the Netherlands, students have a significantly increased chance to kill themselves. By your logic Dutch students are mentally ill.
Transgenders who did not yet undergo treatment can get depressed and it usually happens, this is because of they can't be who they feel they can be which leads to depression.
The reason people who did undergo surgery face a disproportionate amount of discrimination and social exclusion, which leads to depression very fast.
As I said, anyone who is mentally and emotionally stable is allowed to serve and should be allowed to serve. If there are transgenders, queers or travestites, who are emotionally stable and fit to serve, why should they be declined just because they are what they are? Being transgender isn't any more of a mental disorder than beign gay, and I hope to god that we've moved past that point by now.
When it comes to children and transsexualism I think making them undergo treatment is child abuse. Again the studies I've read suggest that 70-80% of children with transsexual feelings and gender confusion stop being confused by the time they're adults. Knowing that I think any doctor who administers puberty delaying drugs to a confused child is guilty of child abuse.
This is a tricky point. Doctors know the statistics. I am not aware of other EU countries but here, Doctors will strongly advise against gender reassignment treatment with underage children. This doesn't mean they can refuse. And your case of child abuse doesn't really fly, considering people have medical self-determination at a younger age than other areas to which the term child abuse applies.