Vegetable oils such as "olive oil, castor oil, or peach kernel oil" are entirely inappropriate for lubricating anything except your salad and steak (or in the case of castor oil, your digestive system).
The problem is that vegetable oils quickly oxidize and get gummy. Think about the residue that accumulates on the bottom and around the cap of a salad oil dispenser. That gummy stuff accumulates dirt and grime and accelerates wear.
The best thing to do is to go to your local auto parts store or hardware store and ask for a good teflon based dry (chain) lube (for the chain), motor oil, or regular auto grease (for the axle). Some people also swear by chainsaw bar oil for bike chains (I find it's kinda sticky and prefer dry lubes, but you might be a rainy area). Here are some lubes that a regular auto/hardware store would have that would suffice:
p.s. There's nothing super special about Phil's -- it's just a good general purpose oil. I actually don't like it, as it is a grit magnet but it has its followers.
FINALLY: Note that your trike is pretty much (sad to say) a Walmart BSO. In many cases, these are very poorly constructed and assembled - so your axle/chain noise may also come from it being poorly constructed and assembled rather than a simple lack of lubrication. If lubing it doesn't work, you may want to try to take it in to a real bike shop for an overhaul -- and because of its construction, it may in fact (even with professional assistance) be impossible to tune to the point where it operates without noise/friction.