You guys are certainly thorough, strange that something is not working, although I think you are focussing here on the bad bits more than the good bits.
Seedlings are best moved just before the first true leaf appears i.e. they have healthy cotyledons, which are the first two thin leaves – or round ones in the case of brassicas, which are robust and the easiest to prick out (cabbage, calabrese, cauliflower etc). I make a hole with a pencil, use the pencil to lift seedlings with 50-100% of their roots, then stuff the roots into the pencil hole, often folded around and with the stem pushed in too so that the seedling is not leggy. It looks rough to an onlooker but as long as you do not break/fracture the soft stem (hold seedlings by leaf), seedlings should sit there for a few days before growing strongly. They do take time to recover, but strangely often grow into plants at least as strong as when directly sown, if not more so.
Water seedlings in after pricking out, then none for two or three days unless it is hot sun. Mine have full daylight in a greenhouse, in a conservatory they should grow fine but will be long stemmed (leggy) from relative lack of light. I have never used artificial light, imagine this helps.
3cm module should be fine for what you describe. Certainly no bigger unless it is peas and beans.