I always use two grafting knives. I use a single beveled felco to cut digonally all the way across the diameter if a scion or rootstock. I like by double beveled opinel carbon 6 or 8 in carbon steel for cleft grafts and cutting the tongues of whip and tongue grafts. It"s all about the right tool for the job.
In High School, we made very serviceable hunting and skinning knives using steel from automotive leaf springs. Our shop teacher told us it was an excellent steel for the purpose. They worked and heat treated easily, sharpened well, and held an edge nicely, although they weren’t capable of the sort of hard, fine edge you might want to shave your face with. For hide, meat, and soft bone they worked well. I can’t recall anyone having any problems with breakage, which was the common complaint with ‘store bought’ knives we’d tried. They might not have been a threat to my Father’s issue model KA-BAR, but held up admirably to tree barks of various types! 
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If you like knives, go to Kershaw’s Amazon website. But do it quickly. They have a sale on Leeks and their Cleaver style in the $7-8 range. I use their cleaver for heavy cutting thick bark grafts. I like the point.
My understanding is that oil is the best so that it doesn’t bend the blade because it cools down a little slower than water. I’m no blacksmith, though I have a friend who is who I could ask. He makes art. https://www.andyalm.com/
It depends on the steel. I know that some steels will fracture in water (learned the hard way) and some won’t harden in an oil quench. But when you’re using road-kill steel you don’t know what you’re working with, although you might be able to guess.
That last time I made a knife I used band saw steel, possibly L6, and I quenched it in water. Don’t know if it was the right thing to do or not!
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There’s a whole lot of ‘it depends.’ Depends on the particular steel, depends on the equipment the maker has available.
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