Kyle
Never assume the battery is in good condition because it is new. Charge it and have it checked. 2, 10 second turns of the key causing a 50% power loss in the battery ain't good. It might not be the battery but seems like an excessive loss.
Builder is correct in his discripion of clearing a flooded engine. But if it is washed with fuel (standing fuel in the intake) you might have to take it a step further as I earlier discribed.
Your leaking carb. is probably due to a bad needle and seat or float and float setting. The needle on the needle and seat has a sealing tip that should never be touched (so they say) by fingers. I have fingered them before and didn't have bad results. I was never able to clean one that was bad and make it work again.
As automarc has pointed out, got the right amount of spark, fuel, compression and the timing is close it will run. It may not run well but it will run.
It is still my best guess as a 37 year vetran in automotive repair, that you have flooded your engine to the point of saturation and it will require more than just holding it to the floor and cranking.
I could be wr-wro-wron mistaken. lol

I didn't do it, no one saw me do it, you can't prove anything! "Bart Simpson"