Specifically, I'd like to find out how to grow ME-3 L.Fermentum (DSM 14241) at home without any fancy equipment. What food does it need? Sugar? Lactose? Something else? At what temperature does it grow? How fast? How can I generally find answers to these questions? ME-3 is not the only strain I'd like to grow. I know ME-3 grows in milk, but that doesn't work for me. Can I grow it in vegan milk? In just water with sugar? I'm even more interested in finding ways to find out the answer to these questions. Where would you look to answer this? I tried to google it but it's hard. Is there not some cool data base that has growth conditions for every known strain?
If you do not have any experience culturing microbes do not attempt this.
You are very very likely to contaminate your culture with things that could make you very sick or even kill you. Pathogenic microorganisms are part of our natural microflora and are a significant cause of hospitalizations throughout the world as a result of food poisoning. For this to work you would need some skills in both identification of microorganisms (to make sure you have the right one) and in sterile technique. Growing specific microorganisms is NOTHING like making yogurt or beer with high inoculums of culture organism that suppress the growth of contaminants, or even like starting a sourdough starter, where you have a relatively low inoculum but the right the culture to the right bacteria/yeasts.
This would mean that you need, at a minimum, a microscope (with 40-100x objectives; for identification purposes), slides, a differential bacterial stain (e.g. Gram stain), sterilization equipment (for preparing media; this could be a pressure cooker), media and an incubator with the right conditions.
This organism needs some carbon, nitrogen and a number of elements to grow happily. I don't think you will be able to do this at home easily; it would require quite a bit of experimentation and you are likely to grow something nasty as a contaminant before you get a decent culture of your target organism.
Proteose Peptone No. 3 10g/L, Beef extract 10g/L, Yeast extract 5g/L, Dextrose 20g/L, Polysorbate 80 1g/L, Ammonium Citrate 2g/L, Sodium Acetate 5g/L, Magnesium Sulfate 0.1g/L, Manganese Sulfate 0.05g/L, Dipotassium Phosphate 2g/L.
Now this might look like a scary list of things, and fairly unobtainable, but beef extract is essentially beef broth, and yeast extract is hydrolyzed yeast. The Polysorbate is an emulsifier used to help solubilize the ingredients and as a carbon source. I don't know if it is obtainable outside of a laboratory setting. The other chemicals may be hard to find too, though citrate = citric acid (fruit acids, you can buy this in a supermarket), acetate = vinegar/acetic acid. Manganese and Magnesium might be obtainable as health supplement pills, but you would need to work out how much is in a pill and do some calculations to convert. As a bonus, supplement pills often contain calcium phosphate as a bulker/tableting agent. For the peptone you might be able to substitute milk powder or "protein shake" powder. You would need to know a bit of chemistry to for the citrate -> citric acid, acetate -> acetic acid, and etc. conversions.
The last problem is that the ATCC recommends that the organism be grown in 5% CO2 at 37 degrees Celsius (98.6 F). This requires a sealed incubator box, with a means to get the CO2 in there. You might be able to achieve this by chemical means, reacting some chalk (calcium carbonate) with acid, but it would take some experimentation to do so effectively. It should grow at lower temperatures too, but it will grow slowly, if at all. I don't know if the CO2 is a requirement or just something that helps with the growth. This is not an obligate anaerobe as suggested in the comments, but can grow anaerobically too (facultative anaerobe).