There are certainly strong correlations between substance abuse and other mental illnesses. However, you have to dig deeper: is the patient an alcoholic because of a preexisting mental illness? Is the alcoholism just the first outward sign that was noticed by others?
There is also a strong correlation among mental illnesses: once you get one (any one), you are statistically more likely to get one or more of the others within your lifetime. It doesn't help that they are often overlapping, though (e.g. bipolar disorder type 2 - bipolar depression). From the American Psychiatric Association Practice Guidelines: "in a study of patients in psychiatric treatment in the United States, 84% of major depressive disorder patients had at least one co-occurring condition: 61% had a co-occurring Axis I condition, 30% a co-occurring Axis II condition, and 58% a co-occurring Axis III condition (978). Anxiety disorders were the most common co-occurring disorder in the prior 12 months" (http://www.psychiatryonline.com/content.aspx?aID=655908)
As a slightly different example, 80-90% of schizophrenic patients smoke. This is because the stimulating effect of nicotine tends to improve their negative symptoms by increasing dopamine levels in certain parts of the brain (for about 15 minutes or so... then they need another one).
As far as alcohol causing depression: alcohol /is/ a depressant. It binds to GABA receptors (among other receptors) and increases sedation, decreases mental activity, etc. (GABA is the primary inhibitory neurotransmitter of the brain).
Inflammation as a cause of depression is not really significant right now. Look at our classes of antidepressants: tricyclics (which work in depression by inhibiting the reuptake of serotonin and norepinephrine), selective serotonin reuptake inhibitors, selective norepinephrine reuptake inhibitors, dopamine reuptake inhibitors (bupropion is mostly what I'm going for here, though sertraline also has some dopaminergic activity), serotonin receptor agonists (e.g. trazodone), and alpha-2 antagonists (mirtazapine - it indirectly increases serotonin and norepinephrine levels by blocking autoinhibition of alpha-2 receptors in the brain). None of these have noteworthy effects on inflammation, and all of them are effective in a modest population of patients (unfortunately only about 1/3 of patients achieve remission on their first drug trial, regardless of which one). Inflammation isn't really a significant theory of depression right now, although I'm not saying it doesn't exist. Mental illnesses are unbelievably complex and medical science is only beginning to catch up with our needs.