On the card titled "Jade Acolyte", it states "spend 1 resource and choose a target development you control. That development becomes a unit with 2 hit points and 1 power until the end of the turn. It also counts as development." So if this "development turned unit" is killed in battle, is the development then placed in your discard pile?

Yes.

If a card is considered both a unit and a development, the rules for both apply to it, and it is affected by effects that affect either. So it is destroyed if it has damage equal to its hit points, and once destroyed it leaves play and enters the discard pile, since that's what units do. Developments can't be damaged, but units can, and card effects don't care what a card is not, only what it is. What else it is doesn't matter. It's a unit, it can be damaged, it is destroyed if it is damaged enough.

Thanks for your help!