It's called the dowsing machine in SoulSilver and HeartGold, you get it from either Ethan/Lyra, or someone you have to do a job for. I can't remember which, I know you have to beat the goldenrod Gym first.
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your rival gives it to you
Yes.
You can find leftovers where snorlax is. Use the item finder at this spot and you will find the leftovers. You cant use the item finder near it you have to beon top of the spot. It will then say "there is an item under foot" and there it will be.
There is no place to buy elixirs in Pokemon Sapphire. You can find them using the item finder.
to get a lucky in soul silver you have to catch a chansy and then take the item if it has a item