Living at home, to a great extent, continues to give them security and moral support. Meanwhile, their parents agree it's ridiculous for their children to sustain an entire rent, and it makes sense for them to stay at home. Meanwhile is a little bit more formal. "At the same time" also works, but if you're writing an essay, it's usually better to use the more formal language. I'd also replace "kids" with "children" for the same reason.

Meanwhile is wrong here. When used to describe time, these are synonyms. However, in your context, you are not describing time. In the discursive sense, 'meanwhile' actually means 'on the other hand'. 'On the other hand' and 'meanwhile' are not appropriate because you are not introducing a counter-argument. You are introducing an additional point. 'At the same time' is perfectly fine. You could also use 'In addition', 'Moreover', etc.

I would say "meanwhile" in this context, no doubt. In an essay avoid contractions, and try some phrasal verbs lol. Anyway, just my 2 coins, after all I am also a student like you ;-) cheers