I know nothing about bonsai, but they are just little trees, so I'll give it a go...
Different plants like different light conditions. Some like shade, some partial shade, some full sun. Some aren't fussy.
However, I think you are on the right track with the soil conditions, Different plants like different soils: moist or free draining, acid or alkali, sandy/sparse or rich and humic. To be certain, you need to know what species the plants are and look them up online.
I'd guess that most trees (even little ones) like a woodland soil. This would be loamy, so equal parts sand, silt and clay plus a decent amount of humic matter, ie. decayed leaf mould. Think, a nice, dark friable woodland soil.
I expect that over-watering and drainage may also be the problem, as you have suggested. Many woodland plants like a moist but free draining soil. Yes, I know, this sounds contradictory, but it's not. It means the plants don't like to fully dry out between watering, but they don't like their roots sitting in wet compost, as it causes root rot.
To combat this, you have to ensure that the pot they are in has plenty of drainage holes (if it's inside, you'll need a pot base/dish too).
Then, and this is important, you need to make sure the pot has lots of drainage at it's base. This could be something like gravel, pea-grit or even coarse bark chips; anything that allows the water from the soil to drain away freely. I've lost count of the half dead plants I've salvaged from back alleys in pots with no drainage holes or no drainage material, just basic used up compost.
The other thing that is useful is feeding the plant/tree every now and again to replenish the nutrients in the potting mix. You can buy the appropriate feed or make your own from seaweed or comfrey. For my inside plants I use dried out banana peel as it is rich in nitrogen.
But really, the first step is identifying your plant's needs online then go from there. I'd hate to give you advice on what to do without knowing the tree's requirements.
