***"1. How can the engine run with a coil that tests open in the secondary circuit?"***
To understand how that can be, you must remember that the secondary circuit of an ignition coil produces voltage ranging from over 20,000 volts to maybe 30,000 volts or more. Voltage like this is called "High Tension" or "High Potential" voltage. Lightning (extremely true) falls into this category). Voltage this high can very easily jump the gap between the broken ends of wiring in the secondary windings of an ignition coil and will produce a somewhat weaker and less defined spark across the plug gap. But the ohmeter you use to check the resistance of the coil might only have a few volts worth of potential (from the meter batttery) and that is not sufficient to jump across the gap between the broken ends of the wire in the secondary coil circuit.

You do not have to use an Onan branded coil. The main criteria that must be met is the resistance value of the primary winding of the coil, and whether the ignition circuit of the engine utilizes an external ballast resistor, or a resistor inside the coil itself. You do not want to use a coil having an internal resistor on an engine that already has an external ballast resistor because the engine will be hard to get started, and if old schoool contact points control the ignition timing, the life of the points will be decreased. If electronic ignition controls the spark timing, the electronic ignition will be overloaded and fail.
Likewise, if you use a coil intended for use with a ballast resistor (non resistor coil), but there is no external ballast resistor in the engine circuit, the life of the coil will be shortened (coil will burn out). If there are any automobile electric specialty shops/stores in your area, they should be able to cross ref a replacement coil for around $50 or less. Take the old coil with you.

Onan engine club at yahoo groups is where you need to be are red tractor dot com. At yahoo group there folks sell used parts you may get used one as low 25 dollars look for Rol Farrar he one I buy used parts from.

In my years of living, i have found that the uninformed who set out to buy something, about which they have no knowlege, and have not looked for info on the product, will be most likely to be "trimmed" by unscrupulous purveyors of the product!
That said, it might be wise to contact somebody who has the smarts about fixing something.
I had a tractor here that had the Onan engine in it, and it had no spark. After much inspection, and checking with smarter folks than I, i managed to borrow a used Onan ignition coil, installed it, and the engine started and ran well. And, i had to go buy a new ignition coil, and returned the used one. This episode was 4 years ago, so prices may be somewhat higher. Near as i can remember it cost $45-50!