Like most ingredients or nutrients, it does not work in isolation. So just giving a child DHA through fish oil/Omega 3s or in small quantities as an additive in a grocery store product yields minimal results. You may see some items on the shelves which are going to have limited effect at high cost, for several reasons. A pre-mixed liquid doesn't have a ton of shelf life, and certain compounds (not all) break down over time. There's also limited absorption in most people - so what you ingest (or your child ingests) is not the same as what is actually absorbed into the cells. Some products (and this goes for vitamins, minerals and many other supplements as well as nutrients in food) don't withstand the effects of digestion or are not fully absorbed for a variety of reasons. One factor is whether a nutrient is eaten with its other necessary "partners" - the interaction of vitamins and minerals, trace elements, and other nutrients is highly complex and most of them don't work without all of the others. That's how nature "provides" nutrition, but with our processed foods, as well as natural foods grown with heavy fertilizers and transported for thousands of miles out of season, and toxins added throughout the growing season or after harvest, the actual make-up of our foods has changed. That's a big part of the reason why, over 25-30 years, we have seen a huge increase in the issues you list and others: ADD, ADHD, Asperger's, autism, sensory processing problems, and more.
DHA provided in full balance with other nutrients to assure higher absorption is very helpful, and we have clinical as well as significant anecdotal evidence to support this. The DHA and the other nutrients have to be high quality, free of contamination, pre-tested and provided with a certificate of analysis to ensure potency and quality, and tested again during and after manufacture. Very few nutrition companies do that, and the FDA does not require it. However, if a company has the FDA Good Manufacturing Practices designation, or if it has a patent on the product (entire formula, not just an ingredient or two), or ideally if it has both, that's a great sign. It's hard to find but not impossible. I can make a referral for you if you like - there is a company that has you tie your order to a nutritional consultant with training to help you. (There is no fee for the consultant - you just pay for the products).
Nothing is an overnight success, so it's important to follow the advice given and try to be as consistent as possible. Results build over time, and it's hard to know how soon a given child will respond. Progress is usually incremental - you may see some improvement in 1-2 weeks, but full results can take longer. Usually you want to make a commitment (at least an "emotional" commitment) of 3 months and then really work with any obstacles to achieve success. For example, there are different ways to get the kids to take it regularly - although the best products are delicious and easy to mix in water or milk or whatever other liquid you give them, or mix into yogurt or pudding. If a child has sensory issues, sometimes there has to be a little subterfuge, but it can easily be overcome with a little work.
St. John's Wort and other ingredients that people will suggest, may have some benefit, but again, hardly ever as a single nutrient.
By and large, single nutrient regimens are not highly successful. Comprehensive plans, with guidance, yield the best results.
This is a food-based approach and does not interfere with any medical or pharmaceutical approach. I'm not surprised by the list you give, and while irritability and sleepiness can be managed, I'm concerned about hearing voices and having hallucinations.
There is significant work being done on epigenetics, which is the "switching" or expression of genes. The genes are inherited, obviously, but the switching (this gene off, this one on) that helps to differentiate cells and their functions (brain cell vs. nerve cell vs. skin cell vs. bone cell and so on which all have the same DNA but which function and look very different from each other) is what is impacted, sometimes severely, by environmental influences (toxins, food additives, pesticides, herbicides, other pollution), trauma (including disease and accidents), nutritional deprivation (because of depleted soils and processed foods), and drug interactions. Tremendous research over the past 16 years has shown there is simple work that can be done using a natural ingredient to restore proper functioning (I think it's difficult to use the word "cure") and boost the potency of a full-spectrum supplement.
I would be happy to work with you more if you like. There is a whole network of parents and also adult sufferers to whom you can be connected for advice, support and great success stories.